GRADINGS ON THE REAL FORMS OF THE ALBERT ALGEBRA, OF 𝔤_{2} AND OF 𝔣_{4}

A. J. CALDERÓN MARTÍN, C. DRAPER FONTANALS, AND C. MARTÍN GONZÁLEZ

Abstract. We find all the fine group gradings on the real forms of the Albert algebra and of the exceptional Lie algebras 𝔤_{2} and 𝔣_{4}.

1. Introduction

A grading on an algebra is a decomposition into the direct sum of subspaces (called homogeneous components) in a way “compatible” with the structure of the algebra. Sometimes the homogeneous components may be decomposed themselves so as to obtain a new grading which is said to be a refinement of the original one. A fine grading is one which can not be further refined. As an instance of a fine grading we can mention the Cartan decomposition of a semisimple complex Lie algebra. The basis corresponding to the Cartan decomposition has been a key piece for the study of the structure and representations of this kind of algebras. But, as pointed out in [25], practically always the bases corresponding to fine gradings (others than the Cartan decomposition) appear naturally in the theory of Lie algebras and in their applications, even though the underlying grading has not been emphasized (e.g. Pauli basis of sl(2, ℂ)).

The interest about gradings from a Lie-theoretic view point began with [27] and [15]. Since then several authors have been working on the problem of the classification of gradings on simple Lie algebras. Within this research line, classical Lie algebras have been studied in [15], [3], [1] and [14], and the exceptional Lie algebras 𝔤_{2} and 𝔣_{4} appear in [5], [2] and [6] (where the case 𝔢_{6} is announced too). Also several papers devote their efforts to 𝔡_{4}, an intermediate case between classical and exceptional (see [7] and [14]). In general the gradings on complex algebras have been intensively studied (a survey including the topic is [20]), but we can not say the same about the real case.

The study of the fine gradings is in part justified by the great amount of their applications. Some of them have been mentioned above, but not others like addition quantum numbers, graded contractions and deformations, and so on. This makes the real case to be of special relevance. The first paper about this topic is [16], which characterizes the so called MAD-groups of the automorphism group of the real form of a classical Lie algebra in terms of the MAD-groups in the complexified algebra, previously found in [15]. While in the complex case there is a bijective correspondence between fine (group) gradings on finite-dimensional simple Lie algebras and MAD-groups, in the real case this is not true. Applications of [16] on concrete Lie algebras are, for instance, [25] and [26], jointly with other works by the same authors. In all of them the authors describe examples of fine (group) gradings using several methods (displayed, fundamental and real basis method) but they do not give a exhaustive list of them. We aboard this problem by giving an exhaustive irredundant list of nonequivalent fine gradings on the real forms of the exceptional complex Lie algebras 𝔤_{2} and 𝔣_{4}. The tool used for 𝔣_{4} can be applied to different Lie algebras, in particular to 𝔤_{2}, although for it we have preferred a more conceptual treatment based on the fact that gradings on octonion algebras (over any ground field) are already known ([8]).

Supported by the Spanish MCYT projects MTM2007-60333, by the UCA project Teoría de Lie y de Espacios de Banach, and by the Junta de Andalucía PAI projects FQM-336, FQM-298, FQM-1215, FQM-2467 and FQM-3737.

The paper is organized as follows. In Section 2 we state some preliminaries and first results on gradings of complex algebras and on the relationship between a grading on a real form L of a complex Lie algebra S with the grading on the algebra S itself. It is also noted that, for a complex semisimple Lie algebra, conjugated quasitori in the automorphism group of the algebra give rise to isomorphic gradings of the same; however, one of the main problems that we have is that it is possible that L inherits the gradings induced by two conjugated quasitori of the automorphism group of S, but the resulting two gradings on L are not isomorphic. We devote Section 3 to the study of this problem in the case of fine gradings, characterizing the number of non-isomorphic gradings on L inherited from all the elements in a conjugation class of a MAD-group Q of the automorphism group of S. This number is related to certain action of the normalizer of Q on the set of conjugations of S whose associated real forms inherit Q. In Section 4 we concrete Section 2 to the case of 𝔣_{4}. In order to know if the real form inherits a determined grading on 𝔣_{4}, we fix representatives of the three real forms and relate their associated conjugations with the automorphisms in the normalizer of a fixed maximal torus. Section 5 contains the main results of the paper. We find all the fine gradings on the real forms of 𝔣_{4}, showing besides that they are always inherited from a fine grading on 𝔣_{4}. In Section 6 we use a different approach to the study of gradings on algebras by considering algebraic group schemes, which can be applied in different settings (see also [20]). As an example of applications we find the fine gradings on the real forms of 𝔤_{2} as a corollary of the gradings on octonions described in [8], and the fine gradings on the real forms of the Albert algebra as a corollary of the results in Section 5. Finally, in Section 7 we give a detailed description of the fine gradings on the real forms of 𝔣_{4}, as well as on the real forms of the Albert algebra. To do that, we make use of constructions of the real forms of the exceptional Lie algebra 𝔣_{4} and of the Albert algebra in terms of symmetric composition algebras given in [10] and [11].

2. Generalities on gradings and real forms

2.1. Basic definitions. Let 𝔏 be a complex or real Lie algebra. The term grading will always mean group grading, that is, a decomposition in vector subspaces 𝔏 = ⊕_{g∈G}𝔏_{g} where G is a finitely generated abelian group and the homogeneous spaces verify [𝔏_{g}, 𝔏_{h}] ⊂ 𝔏_{gh} for any g, h ∈ G (denoting by juxtaposition the product in G). We also assume that G is generated by Supp(G) := {g ∈ G: 𝔏_{g} ≠ 0}, called the support of the grading.

Given two gradings 𝔏 = ⊕_{g∈G}U_{g} and 𝔏′ = ⊕_{h∈H}V_{h} we shall say that they are isomorphic if there are a group isomorphism σ: G → H and an isomorphism φ: 𝔏 → 𝔏′ such that φ(U_{g}) = V_{σ(g)} for all g ∈ G. The above two gradings are said to be equivalent if there are a bijection σ: Supp(G) → Supp(H) and an isomorphism φ: 𝔏 → 𝔏′ such that φ(U_{g}) = V_{σ(g)} for any g ∈ Supp(G).

2.2. Gradings over the complex field. For a finite dimensional complex Lie algebra S we have the following key results:

• Any grading on S is induced by a finitely generated abelian subgroup of diagonalizable automorphisms of the automorphism group of the algebra S. The homogeneous components are the simultaneous eigenspaces relative to the given group of automorphisms.

• To go a little further in the previous idea we need some algebraic groups stuff ([24, §3]). A commutative algebraic group whose identity component is an algebraic torus is called an algebraic quasitorus. An algebraic linear group is a quasitorus if and only if there is a basis relative to which the elements of the quasitorus are simultaneously diagonalizable ([18, p. 99]).

If S = ⊕_{g∈G}S_{g} is a G-grading, for G a finitely generated abelian group, its group of characters 𝔛(G) = hom(G, ℂ^{×}) is a quasitorus and the map ψ: 𝔛(G) → aut_{ℂ}S mapping each α ∈ 𝔛(G) to the automorphism ψ_{α}: S → S given by S_{g} ∋ x ↦ ψ_{α}(x) := α(g)x is a group homomorphism. In particular ψ(𝔛(G)) is a quasitorus of aut_{ℂ}S. Conversely, if Q is a quasitorus of aut_{ℂ}S, Q is formed by semisimple automorphisms and we have a 𝔛(Q)- grading S = ⊕_{g∈𝔛(Q)}S_{g} given by

S_{g} = {x ∈ S : q(x) = g(q)x ∀q ∈ Q}.

• Every quasitorus of aut_{ℂ}S is contained in the normalizer of some maximal torus. This is an algebraic group version of the Borel-Serre theorem for Lie groups (a direct consequence of [28, Theorem 6 and Theorem 3.15, p. 92]).

• A grading on an algebra is said to be fine if it has not proper refinements. A MAD-group of a linear algebraic group is a maximal abelian subgroup of diagonalizable automorphisms. A grading on S is fine if and only if it is induced by a MAD-group of aut_{ℂ}S. Two fine gradings on S are equivalent if and only if the MAD-groups which induce them are conjugated.

• A grading on S is said to be toral if it is produced by automorphisms within a torus of aut_{ℂ}S. In the semisimple case, this happens when there is a Cartan subalgebra 𝔥 contained in the identity homogeneous component of the grading. In such a case the root decomposition of S relative to 𝔥 is a refinement of the given grading.

2.3. Generalities about real forms. For real algebras, we unfortunately do not posses any analogue to the above results. However, it is worth to mention that for a real form L of a classical complex Lie algebra of type A_{n}, B_{n}, C_{n}, or D_{n}, the gradings generated by MAD-groups of aut_{ℝ}L are already fine gradings and that these MAD-groups of aut_{ℝ}L are classified in [16]. But the opposite direction is not true: i.e. not all of the fine gradings on L are generated by MAD-groups of aut_{ℝ}L. There are examples of this fact in [16], to understand them it is enough to think that if there is a ℤ_{n}-grading on L, this is not produced by an order n automorphism of L if n ≠ 2. That is why it is convenient to use squemes techniques, as in Section 6, to deal with the topic of scalar extensions. At the moment we are going to develop new techniques to know when L inherits the grading produced by a MAD-group of aut_{ℂ}L^{ℂ}.

We recall that a real Lie algebra L is called a real form of a complex Lie algebra S if L^{ℂ} = ℂ ⊗_{ℝ} L = L ⊕ iL is isomorphic to S. In such a case, σ: S → S given by σ(x + iy) = x − iy if x, y ∈ L is a conjugation, that is, a conjugate-linear involutive ring automorphism. Conversely, given a conjugation σ of S, the algebra of fixed elements S^{σ} := {x ∈ S : σ(x) = x} is a real form of S. Let σ_{1}, σ_{2} be the conjugations associated to two real forms L_{1}, L_{2} of S, it is said that σ_{1} and σ_{2} are equivalent if L_{1} and L_{2} are isomorphic. This happens if and only if there exists an automorphism ϕ ∈ aut_{ℂ}S such that σ_{2} = ϕσ_{1}ϕ^{−1}.

A real form L of a semisimple complex Lie algebra S is called split if it contains a Cartan subalgebra 𝔥 such that ad_{h} is diagonalizable over ℝ for any h ∈ 𝔥, and it is called compact if its Killing form is definite (necessarily negative). It is well known that any complex semisimple Lie algebra S contains both a split and a compact real form. As in proof of [9, Appendix A, Theorem 1.3], if 𝔥 is a Cartan subalgebra of S, Δ = {α_{i}}_{i=1}^{l} is a system of simple roots relative to 𝔥, and {h_{α_{i}}, e_{α_{i}}, f_{α_{i}} : i = 1, …, l} is a set of canonical generators of S, then we can take B a base of S containing the previous set such that [B, B] ⊂ ℚB. Thus L = ∑_{b∈B} ℝb is a real form of S = L ⊕ iL whose associated conjugation σ_{0} : S → S is determined by σ_{0}(e_{α_{i}}) = e_{α_{i}} and σ_{0}(f_{α_{i}}) = f_{α_{i}} for all i = 1, …, l. If we take now ω ∈ aut_{ℂ}S the automorphism determined by ω(e_{α_{i}}) = −f_{α_{i}} and ω(f_{α_{i}}) = −e_{α_{i}}, then τ := σ_{0}ω is a compact conjugation which commutes with σ_{0}.

We also recall ([9, Appendix A, Theorem 1.11]) that for any complex semisimple Lie algebra S there exists a bijection Ψ between the isomorphism classes of real forms of S and the conjugation classes in aut_{ℂ}S of involutive automorphisms of S. This map is given by Ψ([σ]) = [στ_{σ}], where [·] denotes the corresponding class and τ_{σ} is a compact conjugation that commutes with σ (there always exists such τ_{σ}). We will say that [στ_{σ}] (or στ_{σ}) is a realization of S^{σ}. In particular, ω is a realization of the split real form, and the identity is a realization of the compact real form.

The following result will be very useful in the study of the gradings on compact real forms.

Proposition 1. Let L be a compact real form with a G-grading. Then G is isomorphic to ℤ_{2}^{n} for some n ∈ ℕ.

Proof. Write L = ⊕_{g∈G}L_{g} and fix B a basis of L formed by homogeneous elements. Take some g ∈ G such that 2g ≠ 0 and let us check that g ∉ Supp(G), that is, L_{g} = 0. Take x ∈ L_{g}. For any y ∈ B there is some h ∈ G such that y ∈ L_{h}. Thus ad x(ad x(y)) ∈ L_{2g+h} ≠ L_{h}. Hence, by denoting by κ the Killing form on L, we conclude that κ(x, x) = 0 and, as κ is definite, x = 0. Therefore G is generated by elements of order two. □

2.4. Relationship between the gradings on a complex Lie algebra and the gradings on their real forms. We relate now the gradings on a complex Lie algebra S with the gradings on L, a real form of S.

Definition 1. Let S = ⊕_{g∈G}S_{g} be a grading on a complex Lie algebra and L a real form of S. We say that L inherits the grading on S if L = ⊕_{g∈G}(L ∩ S_{g}). In other words, if there is a grading L = ⊕_{g∈G}L_{g} such that S_{g} = L_{g} ⊕ iL_{g} for any g ∈ G.

If the above grading on S is produced by a quasitorus Q of aut_{ℂ}S we will also say that L inherits the grading induced by Q and such a grading will be called grading on L inherited from Q. It is immediate that if L is determined by a conjugation σ of S, then L = S^{σ} inherits the grading S = ⊕_{g∈G}S_{g} if and only if σ(S_{g}) = S_{g} for any g ∈ G.

Proposition 2. Let S be a complex Lie algebra with a G-grading induced by a quasitorus Q = T P ≤ aut_{ℂ}S whose identity connected component (necessarily a torus) is T, and P is a finite subgroup. If σ: S → S is a conjugation, the real form L = S^{σ} inherits the grading on S if and only if σ commutes with T^{ℝ} := {t ∈ T : Spec (t) ⊂ ℝ} and σpσ^{−1} = p^{−1} for any p ∈ P.

Proof. On one hand, if f is an order n automorphism of S, the grading induced by f is S = ⊕_{i∈ℤ_{n}}S_{i}, where S_{i} is the eigensubspace associated to the eigenvalue ϵ^{i} of f, with ϵ a primitive n-root of the unit. The real form S^{σ} inherits the grading if and only if σ(S_{i}) ⊂ S_{i} for any i = 1 … n. In such a case, for any x ∈ S_{i} we check that fσf(x) = \overline{ϵ^{i}}fσ(x) = \overline{ϵ^{i}}ϵ^{i}σ(x) = σ(x) and hence fσf(x) = σ(x) for each x ∈ S and f^{−1} = σfσ. Conversely, if f^{−1} = σfσ and we take x ∈ S_{i}, then fσ(x) = σf^{−1}(x) = σ((ϵ^{i})^{−1}x) = ϵ^{i}σ(x) so that σ(S_{i}) = S_{i} and S^{σ} inherits the grading on S induced by f.

On the other hand, any ℤ-grading S = ⊕_{n∈ℤ}S_{n} is produced by T = {f_{z} : z ∈ ℂ^{∗}} ≤ aut_{ℂ}S for f_{z}|_{S_{n}} = z^{n}id for any n ∈ ℤ. Of course Spec (f_{z}) = {z^{n} : n ∈ Supp ℤ} so that T^{ℝ} = {f_{α} : α ∈ ℝ^{×}}. Let us check that if S^{σ} inherits this grading, then f_{α}σf_{\frac{1}{α}} = σ for any α ∈ ℝ^{×}. Indeed, if x ∈ S_{n}, then σ(x) ∈ S_{n} and so f_{α}σf_{\frac{1}{α}} (x) = f_{α}σ(\frac{1}{α^{n}} x) = \frac{1}{α^{n}} f_{α}σ(x) = σ(x). Conversely, if f_{α}σf_{\frac{1}{α}} = σ and x ∈ S_{n}, then f_{α}(σ(x)) = α^{n}σ(x) so that σ(x) ∈ S_{n} and S^{σ} inherits the grading.

Finally, combine both results because Q is the direct product of one-dimensional tori and finite abelian groups. □

Throughout the paper, we will make use with no mention of the following fact: For any semisimple complex Lie algebra S, a real form L_{1} = S^{σ} inherits the grading on S induced by a quasitorus Q of aut_{ℂ}S if and only if any other isomorphic real form L_{2} = S^{ϕσϕ^{−1}} (with ϕ ∈ aut_{ℂ}S) inherits the grading on S induced by the quasitorus ϕQϕ^{−1} of aut_{ℂ}S. Furthermore, the gradings on L_{1} and L_{2} are isomorphic.

Definition 2. Let L = ⊕_{g∈G}L_{g} be a grading on the real form L of a complex Lie algebra S. The grading on S given by S = ⊕_{g∈G}S_{g} with S_{g} := L_{g} ⊕ iL_{g} for any g ∈ G, is called the induced grading on S by the grading on L.

Of course any grading on L induces a grading on S in such a way that the grading on L inherited by this grading on S is the primitive grading on L.

Observe that if L inherits a fine grading on S then the grading on L is also a fine grading. However, the grading induced on S by a fine grading on L is not necessarily fine.

Remark 1. In the complex case it is known that conjugated quasitori of the automorphism group of the algebra give rise to isomorphic gradings. However, one of the main problems that we find in the study of the gradings on a real form L of a complex Lie algebra S, is that it may be the case that L inherits the gradings induced by two conjugated quasitori in the automorphism group of S but the resulting two gradings on L are not isomorphic, as the following example shows:

Consider S = sl(2, ℂ) and its basis {h, b, c} with

h = \frac{1}{2} ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , b = \frac{1}{2} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , c = \frac{1}{2} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

So, the multiplication in S is given by [h, b] = c, [h, c] = −b and [b, c] = −h. Consider also its real form L = sl(2, ℝ) = ℝh+ℝb+ℝc. It is clear that L = L_{0} ⊕L_{1} is a ℤ_{2}-grading for

(1)  L_{0} = ℝh, L_{1} = ℝb + ℝc;

and that L = L′_{0} ⊕ L′_{1} is also a ℤ_{2}-grading for

(2)  L′_{0} = ℝb, L′_{1} = ℝh + ℝc.

If (1) and (2) were isomorphic gradings, there would exist φ ∈ aut_{ℝ}L such that φ(h) = αb, φ(b) = βh + γc, φ(c) = β′h + γ′c for certain scalars α, β, γ, β′, γ′ ∈ ℝ. But a map φ defined in this way is automorphism if and only if β′ = −αγ, γ′ = −αβ, β = αγ′, γ = αβ′ and α = βγ′ − γβ′. In particular β′ = −α^{2}β′ and γ′ = −α^{2}γ′ so that β′ = γ′ = 0 and φ can not be bijective, a contradiction. The point is that the induced gradings on S are isomorphic, because this set of equations admits solution in the complex field. In other words, L inherits the gradings induced by the conjugated quasitori Q = ⟨f⟩ ⊂ aut_{ℂ}S and ϕQϕ^{−1} = ⟨ϕfϕ^{−1}⟩, where f is the automorphism given by f(h) = h, f(b) = −b, f(c) = −c and ϕ ∈ aut_{ℂ}S is given by ϕ(h) = ib, ϕ(c) = −ic, ϕ(b) = h, but these gradings are not isomorphic.

The study of the number of non-isomorphic gradings on a real form L inherited from all the elements in a conjugation class of a quasitorus Q of the automorphism group of S is therefore an important question for our purposes. We devote the next section to study this problem in general, which will be applied in Section 5 to the case S = 𝔣_{4}. In this particular case, we will prove that if L inherits the gradings induced by two conjugated MAD-groups of aut_{ℂ}𝔣_{4}, then necessarily the resulting two gradings on L will be isomorphic.

3. The uniqueness of the gradings inherited from conjugations

In this section we characterize the number of non-isomorphic gradings on a real form L inherited from a conjugation class of a MAD-group Q of the automorphism group of a finite-dimensional semisimple complex Lie algebra S, in terms of certain action of the normalizer of Q on the set of conjugations of S whose associated real forms inherit Q. From now on, we will denote N(Q) and Cent(Q) for the normalizer and centralizer of Q in aut_{ℂ}S respectively.

Proposition 3. Let σ_{1}, σ_{2} be two conjugations of S such that the real form S^{σ_{1}} inherits the grading induced by a quasitorus Q of aut_{ℂ}S. Then the real form S^{σ_{2}} inherits the grading induced by Q if and only if there exists h ∈ Cent(Q) such that σ_{2} = σ_{1}h.

Proof. Suppose S^{σ_{2}} inherits the grading induced by Q. By Proposition 2, if we write Q = T P where T is a torus and P a finite abelian subgroup of aut_{ℂ}S, we have pσ_{j} = σ_{j}p^{−1} and tσ_{j}=σ_{j}t for any p ∈ P, t ∈ T^{ℝ} and j = 1, 2. Taking into account these identities, if we define h = σ_{1}σ_{2}, we easily check that ph = hp and th = ht. So h ∈ Cent(Q), being σ_{2} = σ_{1}h.

Conversely, if we take h ∈ Cent(Q), then pσ_{1}h = σ_{1}p^{−1}h = σ_{1}hp^{−1} and tσ_{1}h = σ_{1}th = σ_{1}ht for any p ∈ P and t ∈ T^{ℝ}. And again by Proposition 2, the real form S^{σ_{1}h} inherits the grading induced by Q. □

Proposition 4. Let σ be a conjugation of S such that the real form S^{σ} inherits the fine gradings induced by Q and ϕQϕ^{−1} for some MAD-group Q of aut_{ℂ}S and certain ϕ ∈ aut_{ℂ}S. Then both gradings on S^{σ} are isomorphic if and only if there exists μ ∈ N(Q) such that ϕμ commutes with σ.

Proof. Suppose that both gradings are isomorphic. As the gradings induced by Q and ϕQϕ^{−1} are respectively S = ⊕_{α∈𝔛(Q)}S_{α} and S = ⊕_{α∈𝔛(Q)}ϕ(S_{α}), for S_{α} = {x ∈ S : q(x) = α(q)x for all q ∈ Q}, there exist f ∈ aut_{ℝ}S^{σ} and π ∈ aut𝔛(Q) such that f(S_{α} ∩S^{σ}) = ϕ(S_{π(α)})∩S^{σ}. Hence, the automorphism f̃ ∈ aut_{ℂ}S defined by f̃(x + iy) = f(x) + if(y) if x, y ∈ S^{σ} satisfies f̃σ = σf̃ and f̃(S_{α}) = ϕ(S_{π(α)}). Consider the automorphism μ of S defined by μ = ϕ^{−1}f̃. Since μ(S_{α}) = S_{π(α)}, we have μqμ^{−1}(S_{α}) = S_{α} for any q ∈ Q and hence, μqμ^{−1} is a semisimple element in Cent(Q) and thus in Q. Therefore μ ∈ N(Q) and, as f̃σ = σf̃, then ϕμσ = σϕμ.

Conversely, since μQμ^{−1} = Q we can write S_{α} = {x ∈ S : μqμ^{−1}(x) = α(μqμ^{−1})x for all q ∈ Q} = {x ∈ S : μ^{−1}(x) ∈ S_{π(α)}}, where π(α): Q → ℂ^{×} is defined by π(α)(q) = α(μqμ^{−1}). Taking into account that μ(S_{α}) = S_{π(α)}, we have that f̃:= ϕμ is an automorphism of S satisfying f̃σ = σf̃ and f̃(S_{α}) = ϕ(S_{π(α)}). Finally, it is easy to verify that π: 𝔛(Q) → 𝔛(Q) is a group automorphism and so we conclude that the gradings induced by Q and ϕQϕ^{−1} in S^{σ} are isomorphic. □

Given a MAD-group Q of S, we will denote by ℭ_{Q} the set of conjugations σ of S such that the real form S^{σ} inherits the grading induced by Q. By Proposition 3, if we fix an element σ ∈ ℭ_{Q}, then

ℭ_{Q} = {σq : q ∈ Q and (σq)^{2} = id}.

As S^{μσμ^{−1}} inherits the grading induced by μ^{−1}Qμ = Q for any μ ∈ N(Q), so μσμ^{−1} ∈ ℭ_{Q} and we can consider the action

(3)  \begin{aligned}N(Q) × ℭ_{Q} & → & ℭ_{Q} \\ (μ, ν) & → & μνμ^{−1}. \end{aligned}

Let us see when two gradings inherited from Q are isomorphic. Take σ_{1}, σ_{2} ∈ ℭ_{Q}. If the gradings in S^{σ_{1}} and S^{σ_{2}} inherited from Q are isomorphic it means that there is ϕ ∈ aut_{ℂ}S such that σ_{2} = ϕ^{−1}σ_{1}ϕ and that the gradings on S^{σ_{1}} inherited from Q and ϕQϕ^{−1} are isomorphic. By applying the previous proposition, there is μ ∈ N(Q) such that ϕμσ_{1} = σ_{1}ϕμ, hence μσ_{1}μ^{−1} = σ_{2} so that σ_{1} and σ_{2} are in the same orbit of ℭ_{Q} under the action (3). The converse is also true: if σ_{1} = μσ_{2}μ^{−1} ∈ ℭ_{Q} for certain automorphism μ ∈ N(Q), then the grading on S^{σ_{1}} inherited by Q is obviously isomorphic to the grading on S^{μ^{−1}σ_{1}μ=σ_{2}} inherited by μ^{−1}Qμ = Q.

If we denote by ℭ_{Q,σ} the set of conjugations in ℭ_{Q} equivalent to σ, we can restrict the action (3) to N(Q)×ℭ_{Q,σ} → ℭ_{Q,σ} and reformulate our previous ideas as follows:

Proposition 5. Let σ be a conjugation of S such that the real form S^{σ} inherits the grading induced by a MAD-group Q of aut_{ℂ}S. Then the number of non-isomorphic gradings on S^{σ} inherited from the MAD-groups of aut_{ℂ}S conjugated to Q is equal to the number of orbits in ℭ_{Q,σ} under the action (3).

Remark 2. In relation with the number of non-isomorphic gradings on a real form L inherited from a conjugation class of a quasitorus Q which is not a MAD-group of aut_{ℂ}S, we have not considered this possibility here because for the real forms of S ∈ {𝔤_{2}, 𝔣_{4}} there is no fine grading whose induced grading on S is not fine. This will be a consequence of Proposition 6 and Theorem 5.

4. Preliminaries on the gradings and the real forms of 𝔣_{4}

4.1. MAD-groups of aut_{ℂ}𝔣_{4}. We begin by fixing some notations and stating some results concerning 𝔣_{4} from [6]. In fact, this will be the key reference in the present section.

The most known model of 𝔣_{4}, the 52-dimensional complex exceptional simple Lie algebra, is due to Chevalley and Schafer [4], who showed that 𝔣_{4} is the algebra of derivations of the Albert algebra J, the only exceptional simple Jordan algebra under our hipothesys on the field. An interesting fact is that the automorphism group of 𝔣_{4} is isomorphic to F_{4} the automorphism group of J [19, p. 285]. Its Dynkin diagram is

∘  ∘  ∘ >  ∘ α_{4} α_{2}  α_{3} α_{1}

We can choose B a ℂ-basis of 𝔣_{4} satisfying [B, B] ⊂ ∑_{b∈B} ℚb, as in any simple Lie algebra. Of course this choice of basis is not unique: in [6, Section 4] there is a fixed basis, which will be our standard basis B of 𝔣_{4} from now on, in order to work with the automorphisms of 𝔣_{4} by means of their matrices relative to B. Although we are not going to provide here the detailed description of each element, we can precise a little bit that B = {t_{α_{i}}, b_{j} : i = 1, …, 4, j = 1, …, 48} where all the elements b_{j}’s are root vectors relative to the Cartan subalgebra 𝔥 = ⨁_{i=1}^{4} ℂt_{α_{i}}, the element t_{α} is determined by κ(t_{α}, ·) = α ∈ 𝔥^{∗} for κ the Killing form, and one set of canonical generators is

(4)  \begin{matrix}e_{α_{1}} = 2b_{4} & f_{α_{1}} = 2b_{28} & h_{α_{1}} = t_{α_{1}} \\ e_{α_{2}} = 2b_{3} & f_{α_{2}} = −2b_{27} & h_{α_{2}} = t_{α_{2}} \\ e_{α_{3}} = 4b_{2} & f_{α_{3}} = −4b_{26} & h_{α_{3}} = 2t_{α_{3}} \\ e_{α_{4}} = 4b_{1} & f_{α_{4}} = −4b_{25} & h_{α_{4}} = 2t_{α_{4}} \end{matrix}

In fact the b_{i}’s have been chosen as certain elements [R_{x}, R_{y}] with x, y in a standard basis of the Albert algebra J (R_{x} the right multiplications operator), where such basis of J was taken from the standard basis of the octonions. The precise choice of B is necessary if one wants to have the matrices related to the automorphisms (these matrices can be downloaded in [23]), but we will not use them for the description of the gradings on each of the real forms (see Section 7), we will only use them for the technical proofs of the results, in Section 5.

All the fine gradings on 𝔣_{4} have been described in [6] as the simultaneous diagonalization relative to the MAD-groups of the group of automorphisms of 𝔣_{4}. These quasitori can be found (up to conjugation) living in the normalizer of a maximal torus of the automorphism group. We have fixed the maximal torus 𝔗 which acts as the identity in the Cartan subalgebra 𝔥, so that we call t_{x,y,z,u} the element in 𝔗 which acts with eigenvalues x, y, z, u ∈ ℂ^{∗} in the root spaces corresponding to the roots {α_{4}, −α_{1} − α_{2} − 2α_{3} − α_{4}, α_{1} + 2α_{2} + 2α_{3} + 2α_{4}, −α_{3}} respectively (we have not chosen the simple roots {α_{i}}_{i=1}^{4} in order to use the package in [23], which in [6] was taken for a joint treatment of the Albert algebra and 𝔣_{4}). In other words, its matrix relative to B is:

t_{xyzu} := diag(1, 1, 1, 1, x, \frac{1}{u}, y z, \frac{u^{2}}{x y^{2} z} , \frac{x}{u}, \frac{y z}{u} , \frac{u^{2}}{x y} , \frac{u}{x y} , \frac{x y z}{u} , \frac{y z}{u^{2}} , \frac{u}{y}, \frac{x y z}{u^{2}} , \frac{1}{x y} , \frac{1}{y} ,

\frac{x^{2} y z}{u^{2}} , \frac{z}{x}, z, \frac{x}{y}, \frac{z}{u}, x z, \frac{x z}{u} , \frac{x z}{u^{2}} , \frac{x y z^{2}}{u^{2}} , \frac{z}{y} , \frac{1}{x}, u, \frac{1}{y z} , \frac{x y^{2} z}{u^{2}} , \frac{u}{x}, \frac{u}{y z} , \frac{x y}{u^{2}} ,

\frac{x y}{u} , \frac{u}{x y z} , \frac{u^{2}}{y z} , \frac{y}{u}, \frac{u^{2}}{x y z} , x y, y, \frac{u^{2}}{x^{2} y z} , \frac{x}{z}, \frac{1}{z}, \frac{y}{x}, \frac{u}{z}, \frac{1}{x z} , \frac{u}{x z} , \frac{u^{2}}{x z} , \frac{u^{2}}{x y z^{2}} , \frac{y}{z} ), (5)

As there is a short exact sequence 1 → 𝔗 → N(𝔗) →^{π} W → 1, for W the Weyl group of 𝔣_{4}, the normalizer of the torus can be described by means of a copy of W and the torus 𝔗, although we have to take into account that the sequence does not split, that is, there is not a cross section: we cannot define W → N(𝔗) a group monomorphism. But we can take without problems an element in π^{−1}(σ) for each σ ∈ W, according to the extension theorems ([17, §14]). In general we can choose an automorphism sending t_{α_{i}} to t_{σ(α_{i})} and x_{α_{i}} to x′_{σ(α_{i})} for any α_{i} simple root (with x_{α_{i}}, x′_{α_{i}} ∈ (𝔣_{4})_{α_{i}} arbitrarily chosen). If we take x_{α_{i}}, x′_{α_{i}} ∈ B, the automorphism is

then determined, call it σ̄, and its matrix relative to B is block diagonal (\begin{matrix}A & 0 \\ 0 & D \end{matrix})

where A is just the 4×4 matrix of σ relative to the basis {α_{i}}_{i=1}^{4} and D is a 48×48 matrix with only one nonzero element in each row and in each column. Besides D turns out to have entries in {±1}. Thus,

N(𝔗) = {σ̄t : σ ∈ W, t ∈ 𝔗}.

Although the injection W → N(𝔗), σ ↦ σ̄ is well defined, it does not coincide exactly with the injection σ ↦ σ̃ which we will use through this paper, but σ̃ = σ̄t_{σ} for certain t_{σ} ∈ T = {t_{x,y,z,u} : x^{2} = y^{2} = z^{2} = u^{2} = 1}, that is, σ̃ and σ̄ agree up to some signs. This fact has no relevance in [6], in which the authors don’t precise the choice they take for σ̃ because taking σ̄ instead of σ̃ wouldn’t modify the result. But in our work of real forms the choice is quite relevant because an involutive automorphism σ̄t with t ∈ T, will be associated (by means of ψ^{−1}, see Subsection 2.3) to a conjugation of split, compact or nonsplit and noncompact type, depending on the element t. The reason why we do not use here the extension σ̄ is computational: although they have been defined easily, the computation is quite tedious. Instead of that, the procedure used to get the package [23] has been the following: we made W̃ the group generated by {σ̄_{α_{i}} : i = 1, … 4}, which turned out to be 4 copies of the group W (we knew that W̃ could not be one only copy of W). As the software Mathematica orders the elements lexicographically, if σ_{i} is the ith element in W (also with lexicographical order), just the elements in the positions 4i − 3, 4i − 2, 4i − 1, 4i have in the left upper corner the matrix of σ_{i} relative to {α_{i}}_{i=1}^{4}, so we have taken σ̃_{i} ∈ π^{−1}(σ_{i}) to be the 4ith element in W̃.

We recall that, following Subsection 2.2, any fine grading on 𝔣_{4} is induced by a MAD-group of aut_{ℂ}𝔣_{4}, which can be found inside the normalizer N(𝔗). These MAD-groups are computed in [6, Sections 4 and 5]. If we denote by 𝔗^{<j>} the elements in 𝔗 commuting with σ̃_{j} and by Q(j, t) the quasitorus generated by 𝔗^{<j>} and σ̃_{j}, then, every MAD-group is conjugated to one of the following:

10 A. J. CALDERÓN MARTÍN, C. DRAPER FONTANALS, AND C. MARTÍN GONZÁLEZ

Q_{1} := Q(748, id) = 𝔗 ≅ (ℂ^{×})^{4},
Q_{2} := Q(405, id) = ⟨{σ̃_{405}, t_{xyzu} : x^{2} = y^{2} = t^{2} = u^{2} = 1}⟩ ≅ ℤ_{2}^{5},
Q_{3} := Q(105, id) = ⟨σ̃_{105}, t_{111u}, t_{1−1−11}, t_{−11−11} : u ∈ ℂ^{∗}⟩ ≅ (ℂ^{×}) × ℤ_{2}^{3},
Q_{4} := Q(15, id) = ⟨σ̃_{15}, t_{1ϵϵ1}, t_{ϵ1ϵ^{2}ϵ^{2}}⟩ ≅ ℤ_{3}^{3}

(with ϵ a primitive cubic root of the unit), which induce respectively a ℤ^{4}, ℤ_{2}^{5}, ℤ_{2}^{3} × ℤ, and ℤ_{3}^{3}-grading on 𝔣_{4}. We will call Q_{1}, Q_{2}, Q_{3} and Q_{4} the standard MADgroups of aut_{ℂ}𝔣_{4} and the gradings on 𝔣_{4} induced by them the standard fine gradings on 𝔣_{4}.

4.2. About the three real forms of 𝔣_{4}. Recall as in Subsection 2.3 that there exists a bijection Ψ between the isomorphism classes of real forms of 𝔣_{4} and the conjugation classes in aut_{ℂ}𝔣_{4} of involutive automorphisms of 𝔣_{4}. There are two involutive automorphisms of 𝔣_{4} different from the identity. One of them fixes a subalgebra of type 𝔟_{4} (we say that such automorphism f is of type 2A or that f ∈ 2A) and the other one fixes a subalgebra of type 𝔠_{3} ⊕𝔞_{1} (automorphism of type 2B). In consequence 𝔣_{4} has three isomorphisms classes of real forms, determined by the dimension of the fixed subalgebra of its image under Ψ, respectively 52, 36 and 24. Also the isomorphy class of a real form is characterized by the signature of its Killing form, which in the previous cases becomes −52, −20 and 4 respectively.

Taking into account that given two isomorphic real forms L_{1} and L_{2}, any grading on L_{1} gives rise to an isomorphic grading on L_{2}, it is enough to find the fine gradings just on one of the elements in each of the isomorphism classes of real forms. So, we are going to fix one comfortable representative real form in any of the three isomorphism classes of real forms on 𝔣_{4}.

First we fix some notations coherent with Subsection 2.3. We denote by σ_{0} : 𝔣_{4} → 𝔣_{4} the conjugation determined by σ_{0}(e_{α_{i}}) = e_{α_{i}} and σ_{0}(f_{α_{i}}) = f_{α_{i}} for all i = 1, …, 4, where e_{α_{i}}, f_{α_{i}} are described in (4) in terms of B. Take also ω ∈ aut_{ℂ}𝔣_{4} the involutive automorphism determined by ω(e_{α_{i}}) = −f_{α_{i}} and ω(f_{α_{i}}) = −e_{α_{i}}, and τ := σ_{0}ω the compact conjugation which commutes with σ_{0}. Note that ω maps (𝔣_{4})_{α} into (𝔣_{4})_{−α}, hence π(ω) = −id_{4} = σ_{405} ∈ W. A priori, ω differs from any extension of σ_{405} in an order two element in the torus, but the precise choice of σ̃_{405} in [6] was the automorphism mapping b_{1}, b_{2}, b_{3}, b_{4} into b_{25}, b_{26}, b_{27}, −b_{28} respectively, that is, just ω = σ̃_{405}. Clearly, for the split conjugation σ_{0} we have Ψ([σ_{0}]) = [σ_{0}τ] = [ω] and dim Fix(ω) = 24; and for the compact conjugation τ we have Ψ([τ]) = [id_{𝔣_{4}}] and dim Fix(id) = 52. A conjugation in the last class can be taken θ := τt_{111−1}, because τ commutes with t_{1,1,1,−1} so that Ψ([θ]) = [t_{1,1,1,−1}] but dim Fix(t_{1,1,1,−1}) = 36.

From now on,

(1) 𝔣_{4,4} will denote the standard split real form determined by the conjugation σ_{0}.

(2) 𝔣_{4,−52} will denote the standard compact real form determined by the conjugation τ = σ_{0}ω.

(3) 𝔣_{4,−20} will denote the standard nonsplit and noncompact real form determined by the conjugation θ = τt_{111−1}.

Relative to these classes, if Q is a MAD-group of 𝔣_{4}, we will also denote by ℭ_{Q,i} with i = 1, 2, 3 the set ℭ_{Q,σ} (possibly empty) for σ a conjugation in ℭ_{Q} of split, compact or nonsplit and noncompact type, respectively. Thus, ℭ_{Q} is the disjoint union ℭ_{Q} = ℭ_{Q,1} ∪ ℭ_{Q,2} ∪ ℭ_{Q,3}.

4.3. Relationship between the normalizer of the torus and the standard conjugations. We are going to state the relations between the elements of the normalizer of our torus, in particular the elements in the standard MAD-groups Q_{1}, Q_{2}, Q_{3} and Q_{4} inducing the standard fine gradings on 𝔣_{4}, and the conjugations σ_{0}, τ and θ which determine the standard real forms:

Lemma 1. The following assertions hold for any x, y, z, u ∈ ℂ^{∗} and for any i ∈ {1, ..., 1152} (|W | = 1152),

\begin{matrix}(1) σ_{0}σ̃_{i} = σ̃_{i}σ_{0}, \\ σ_{0}t_{xyzu} = t_{\overline{x} \overline{y} \overline{z} \overline{u}}σ_{0}. \end{matrix}

(2) ωσ_{0} = σ_{0}ω,

ωσ̃_{i} = σ̃_{i}ω,
ωt_{xyzu} = t_{x^{−1}y^{−1}z^{−1}u^{−1}}ω.

(3) τσ_{0} = σ_{0}τ,

τσ̃_{i} = σ̃_{i}τ,
τt_{xyzu} = t\overline{x}^{−1}\overline{y}^{−1}\overline{z}^{−1}\overline{u}^{−1}τ.

Proof. (1) On one hand, as σ_{0} is defined to be the identity in B′ := {b_{1}, b_{2}, b_{3}, b_{4}, b_{25}, b_{26}, b_{27}, b_{28}}, and every element in B belongs to the ℚ-algebra generated by B′, then σ_{0} acts also as the identity on every element in B. But, if j = 1, 2, 3, 4, the element σ̃_{i}(b_{j}) belongs to ℚB (in fact, is the only one element in B ∩ (𝔣_{4})_{σ_{i}(α_{4−j})} up to the sign) so that σ_{0}σ̃_{i}σ_{0}(b_{j}) = σ_{0}σ̃_{i}(b_{j}) = σ̃_{i}(b_{j}) and we conclude that the automorphisms σ_{0}σ̃_{i}σ_{0} and σ̃_{i} agree. On the other hand σ_{0}t_{xyzu}σ_{0} = t_{\overline{x} \overline{y} \overline{z} \overline{u}} because σ_{0} is conjugate-linear so that both automorphisms act on the same way on B′ (see equation (5)).

(2) It is known from [23] that σ̃_{405} commutes with all our extensions σ̃_{i}. It is also clear ωt_{xyzu} = t_{x^{−1}y^{−1}z^{−1}u^{−1}}ω because ωtω^{−1} = π(ω) · t = −id_{W} · t for · the action of the Weyl group on the torus given in [6, p. 870].

(3) As τ = σ_{0}ω, it is an immediate consequence of (1) and (2). □

From now on, we will make use of Lemma 1 with no explicit reference to it.

5. Fine gradings on the real forms of 𝔣_{4} and on the Albert algebra

To study the fine gradings on the real forms of 𝔣_{4} we will follow the next program. In a first step we will verify which of the fine gradings on 𝔣_{4} are inherited by some of the standard real forms 𝔣_{4,4}, 𝔣_{4,−52} and 𝔣_{4,−20} and if they yield several isomorphy classes. In a second step we will study if these real forms inherit some other grading, not necessarily fine, on 𝔣_{4}.

5.1. Fine gradings inherited from the conjugation class of Q_{1}. We recall that Q_{1} = 𝔗 induces on 𝔣_{4} a fine toral ℤ^{4}-grading, just the Cartan decomposition of this algebra relative to 𝔥. As σ_{0} commutes with 𝔗^{ℝ} by Lemma 1, Proposition 2 implies that σ_{0} ∈ ℭ_{Q_{1}} and that the standard split real form 𝔣_{4,4} inherits the grading induced by 𝔗. Of course, a real form of 𝔣_{4} which inherits this grading is necessarily split, that is, ℭ_{Q_{1}} = ℭ_{Q_{1},1}. The uniqueness of the gradings inherited from conjugations is clear in this case because all the brackets of root vectors can be recovered from the Cartan matrix of 𝔣_{4}.

Let us use Section 2 techniques to check again that there is only one orbit in ℭ_{Q_{1}} under the action of N(Q_{1}) (in particular, ℭ_{Q_{1},2} = ℭ_{Q_{1},3} = ∅). First note that the elements in ℭ_{Q_{1}} are σ_{0}t, for any t ∈ 𝔗 such that (σ_{0}t)^{2} = id, which is equivalent to tt̄= id (by denoting t̄_{xyzu} = t_{\overline{x} \overline{y} \overline{z} \overline{u}}). Hence

ℭ_{Q_{1}} = {σ_{0}t : t ∈ 𝔗^{S^{1}}},

where we are denoting by S^{1} = {z ∈ ℂ : |z| = 1} and 𝔗^{S^{1}} := {t ∈ 𝔗 : Spec (t) ⊂ S^{1}} = {t_{a_{1}a_{2}a_{3}a_{4}} : a_{i} ∈ S^{1}, i = 1, 2, 3, 4}. If t ∈ 𝔗^{S^{1}}, there are α_{j} ∈ ℝ such that t = t_{a_{1}a_{2}a_{3}a_{4}} for a_{j} = e^{2πiα_{j}}. By taking μ = t_{a′_{1}a′_{2}a′_{3}a′_{4}} ∈ Q_{1} ⊂ N(Q_{1}) with a′_{j} = e^{−πiα_{j}} ∈ S^{1}, we have μ̄μ^{−1} = t and hence μσ_{0}μ^{−1} = σ_{0}t. Therefore every element in ℭ_{Q_{1}} is in the same orbit (the orbit of σ_{0}). In particular the realization Ψ([σ_{0}t]) = [ωt] is of type 2B and dim Fix(ωt) = 24 for any t ∈ 𝔗^{S^{1}}.

5.2. Fine gradings inherited from the conjugation class of Q_{2}. The conjugations σ_{0} and τ commute with every element in Q_{2} by Lemma 1, and so does θ = τt_{111−1}. This implies, by Proposition 2, that σ_{0}, τ, θ ∈ ℭ_{Q_{2}} and that the standard real forms 𝔣_{4,4}, 𝔣_{4,−52} and 𝔣_{4,−20} inherit the fine grading on 𝔣_{4} induced by Q_{2}.

Let us study the problem of the uniqueness in this case. As σ_{0} commutes with Q_{2},

ℭ_{Q_{2}} = {σ_{0}q : q ∈ Q_{2} and (σ_{0}q)^{2} = id} = σ_{0}Q_{2}.

In order to know the isomorphy class of a conjugation σ in ℭ_{Q_{2}}, it is enough to compute the dimension of the subalgebra fixed by a representative of Ψ([σ]). As τ commutes with any conjugation in ℭ_{Q_{2}}, we get that for any t ∈ 𝔗^{<405>}, Ψ([σ_{0}t]) = [σ_{0}tτ] = [ωt] and so σ_{0}t ∈ ℭ_{Q_{2},1} (see previous subsection), and that Ψ([σ_{0}ωt]) = [t]. But dim Fix(t) = 52 if t = id, dim Fix(t) = 36 if t ∈ 2A_{𝔗} := 2A ∩ 𝔗 = {t_{111−1}, t_{−1−1−11}, t_{−1−1−1−1}}, and dim Fix(t) = 24 if t ∈ 2B_{𝔗} := 2B ∩ 𝔗, that is, in the remaining twelve cases, so that we can write

ℭ_{Q_{2},1} = σ_{0}𝔗^{<405>} ∪ {σ_{0}ωt : t ∈ 2A_{𝔗}},
ℭ_{Q_{2},2} = {σ_{0}ω}, and
ℭ_{Q_{2},3} = {σ_{0}ωt : t ∈ 2B_{𝔗}}.

As a consequence ℭ_{Q_{2},1} has 28 elements, ℭ_{Q_{2},2} one element and ℭ_{Q_{2},3} three elements. We have to study the number of orbits in each ℭ_{Q_{2},i} under the action of N(Q_{2}).

We begin by considering the split case. We will show that there is only one orbit in ℭ_{Q_{2},1}. First, we note that every element in σ_{0}𝔗^{<405>} is contained in the same orbit. This is a consequence of the fact that given any t ∈ 𝔗^{<405>}, we can take μ ∈ 𝔗 such that μ^{2} = t, so that μ^{4} = id and μσ_{0}μ^{−1} = σ_{0}μ^{−2} = σ_{0}t, where such element μ belongs to N(Q_{2}) (μσ̃_{405}μ^{−1} = σ̃_{405}μ^{−2} ∈ Q_{2}). Second, we are checking that {σ_{0}ωt : t ∈ 2A_{𝔗}} is also contained in a same orbit under our action. Indeed, it is well known that two elements in a maximal torus conjugated in aut_{ℂ}𝔣_{4} are also conjugated in the normalizer of the torus, so that given t_{1}, t_{2} ∈ 2A_{𝔗}, there exists j ∈ {1, ..., 1152} such that σ̃_{j}t_{1}σ̃_{j}^{−1} = t_{2}. By taking μ = σ̃_{j}, Lemma 1 gives us μσ_{0}ωt_{1}μ^{−1} = σ_{0}ωt_{2}. But μ ∈ N(Q_{2}) because μσ̃_{405}μ^{−1} = σ̃_{405} and for any t ∈ 𝔗^{<405>}, μtμ^{−1} is an order two element in 𝔗, hence belonging to 𝔗^{<405>}. Finally, let us prove that σ_{0} and σ_{0}ωt belong to the same orbit under the above action for some t ∈ 2A_{T}. By denoting t_{1} = t_{−1111}, t_{2} = t_{1−111}, t_{3} = t_{−1−1−11} and t_{4} = t_{111−1}, we can write Q_{2} = ⟨t_{1}, t_{2}, t_{3}, t_{4}, ω⟩. Since the subquasitorus ⟨t_{1}, t_{3}, t_{4}, ω⟩ is toral (because it fixes a four dimensional abelian subalgebra), [6, Proposition 7] gives us that there exists p ∈ aut_{ℂ}𝔣_{4} such that pQ_{2}p^{−1} ⊂ N(𝔗), p⟨t_{1}, t_{3}, t_{4}, ω⟩p^{−1} ⊂ 𝔗 and pt_{i}p^{−1} = t_{i} for i = 1, 3, 4. As p⟨t_{1}, t_{3}, t_{4}, ω⟩p^{−1} is isomorphic to ℤ_{2}^{4}, then necessarily p⟨t_{1}, t_{3}, t_{4}, ω⟩p^{−1} = 𝔗^{<405>}. Since pt_{2}p^{−1} ∈ N(𝔗) commutes with the whole 𝔗^{<405>}, then π(pt_{2}p^{−1}) ∈ {σ_{405}, id} and we conclude pt_{2}p^{−1} = ωt with t ∈ 𝔗 (if pt_{2}p^{−1} ∈ 𝔗, then pQ_{2}p^{−1} ⊂ 𝔗, which is an absurd). Let us take μ := sp where s ∈ 𝔗 such that s^{−2} = t. Taking into account Lemma 1 and the previous facts pt_{i}p^{−1} = t_{i} for i = 1, 3, 4, pt_{2}p^{−1} = ωt and pωp^{−1} ∈ 𝔗^{<405>}, we can assert that μt_{i}μ^{−1} = t_{i} for i = 1, 3, 4, μt_{2}μ^{−1} = ω and μωμ^{−1} ∈ 𝔗^{<405>} \ ⟨t_{1}, t_{3}, t_{4}⟩. In particular, μQ_{2}μ^{−1} = Q_{2}, that is μ ∈ N(Q_{2}).

If μσ_{0}μ^{−1} ∈ {σ_{0}ωt : t ∈ 2A_{𝔗}} we have finished. Otherwise μσ_{0}μ^{−1} ∈ ℭ_{Q_{2},1} \ {σ_{0}ωt : t ∈ 2A_{𝔗}} = σ_{0}𝔗^{<405>} and μσ_{0}ωt_{1}μ^{−1} = (μσ_{0}μ^{−1})(μωμ^{−1})t_{1} ∈ σ_{0}𝔗^{<405>} so that we have finished too. As a consequence every element in ℭ_{Q_{2},1} is in the same orbit and, according to Proposition 5, if 𝔣_{4,4} inherits the grading induced by some quasitorus conjugated to Q_{2}, then the resulting grading is isomorphic to the one inherited from Q_{2}.

It is clear that ℭ_{Q_{2},2} has an only orbit with an only element.

Finally consider the set ℭ_{Q_{2},3} = {σ_{0}ωt : t ∈ 2B_{𝔗}}. If t_{1}, t_{2} ∈ 2B_{𝔗}, we can argue as in the above split case to get that there exists j ∈ {1, ..., 1152} such that σ̃_{j}t_{1}σ̃_{j}^{−1} = t_{2}. Hence μσ_{0}ωt_{1}μ^{−1} = σ_{0}ωt_{2} for μ = σ̃_{j}, which belongs to N(Q_{2}). Consequently, the action N(Q_{2}) × ℭ_{Q_{2},3} → ℭ_{Q_{2},3} has only one orbit, and, in case 𝔣_{4,−20} inherits the grading induced by some quasitorus conjugated to Q_{2}, the inherited grading is isomorphic to the one inherited from Q_{2}.

5.3. Fine gradings inherited from the conjugation class of Q_{3}. Lemma 1-(1) gives us that σ_{0} commutes with 𝔗^{<105>} ∩ 𝔗^{ℝ} and with σ̃_{105}(𝔗^{<105>} ∩ 𝔗^{ℝ}). Thus Proposition 2 implies that σ_{0} ∈ ℭ_{Q_{3}} and that 𝔣_{4,4} inherits the fine grading on 𝔣_{4} induced by Q_{3}.

As Q_{3} induces a ℤ_{2}^{3} × ℤ-grading, Proposition 1 ensures that any compact real form does not inherit neither the fine grading on 𝔣_{4} induced by Q_{3} nor, by the same reason, another fine grading induced by a quasitorus conjugated with Q_{3}. Thus, ℭ_{Q_{3},2} = ∅.

Finally, let us study the case concerning the standard real form 𝔣_{4,−20} determined by θ = τt_{111−1}. Observe that 𝔣_{4,−20} does not inherit the fine grading on 𝔣_{4} induced by Q_{3} because t_{111u} only commutes with θ if u = ū^{−1}, not for any u ∈ ℝ^{×}. However, it is possible that 𝔣_{4,−20} inherits the fine grading on 𝔣_{4} induced by a MAD-group conjugated of Q_{3}, or equivalently, that some real form isomorphic to 𝔣_{4,−20} inherits the fine grading on 𝔣_{4} induced by Q_{3}. In fact, if we consider the conjugation σ_{0}σ̃_{105}, with realization Ψ([σ_{0}σ̃_{105}]) = [σ̃_{405}σ̃_{105}], then it is a straightforward computation that dim Fix(σ̃_{405}σ̃_{105}) = 36. And besides this conjugation σ_{0}σ̃_{105} ∈ ℭ_{Q_{3}} because it commutes with 𝔗^{<105>} ∩ 𝔗^{ℝ} and with σ̃_{105}. That is, S^{σ_{0}σ̃_{105}} inherits the fine grading on 𝔣_{4} induced by Q_{3} and so 𝔣_{4,−20} inherits the fine grading induced by certain MAD-group conjugated to Q_{3}.

We aboard now the problem of the uniqueness, following the same program as in the previous two subsections. As σ_{0} ∈ ℭ_{Q_{3}}, we can write

ℭ_{Q_{3}} = {σ_{0}q : q ∈ Q_{3} and (σ_{0}q)^{2} = id}.

If t ∈ 𝔗^{<105>}, then (σ_{0}t)^{2} = tt̄ = (σ_{0}σ̃_{105}t)^{2}, so σ_{0} and σ_{0}σ̃_{105} have order two if t ∈ 𝔗^{S^{1}}. So we can write

ℭ_{Q_{3}} = {σ_{0}t, σ_{0}σ̃_{105}t : t ∈ 𝔗^{<105>} ∩ 𝔗^{S^{1}}}.

As t ∈ 𝔗^{<105>} ∩𝔗^{S^{1}} implies τt = tτ, we get easily that Ψ([σ_{0}t]) = [σ_{0}tτ] = [ωt] and Ψ([σ_{0}σ̃_{105}t]) = [σ̃_{405}σ̃_{105}t]. Now we know that dim Fix(ωt) = 24 (Subsection 5.1) and we compute dim Fix(σ̃_{405}σ̃_{105}t_{x,y,xy,u}), which turns out to be 36 if x = y = 1 and 24 if either x = −1 or y = −1. Hence, we can assert

ℭ_{Q_{3},1} = σ_{0}(𝔗^{<105>} ∩ 𝔗^{S^{1}}) ∪ σ_{0}σ̃_{105}{t_{−11−1u}, t_{1−1−1u}, t_{−1−11u} : u ∈ S^{1}}, ℭ_{Q_{3},2} = ∅, ℭ_{Q_{3},3} = {σ_{0}σ̃_{105}t_{111u} : u ∈ S^{1}}.

We begin by considering the split case. We will show that there is just one orbit of ℭ_{Q_{3},1} under the action N(Q_{3})×ℭ_{Q_{3},1} → ℭ_{Q_{3},1}. First, observe that σ_{0}(𝔗^{<105>}∩𝔗^{S^{1}}) is contained in one orbit. Indeed, if t ∈ 𝔗^{<105>} ∩ 𝔗^{S^{1}}, we have t = t_{a_{1}a_{2}a_{3}a_{4}} for a_{j} = e^{2πiα_{j}} with a_{1}^{2} = a_{2}^{2} = 1, a_{3} = a_{1}a_{2} and a_{4} ∈ S^{1}. By taking as in 5.1 the element μ = t_{a′_{1}a′_{2}a′_{3}a′_{4}} with a′_{j} = e^{−πiα_{j}} ∈ S^{1}, we have μσ_{0}μ^{−1} = σ_{0}t. Furthermore, μ ∈ N(Q_{3}) since μσ̃_{105}μ^{−1} = t_{\frac{1}{a_{1}} \frac{1}{a_{2}} \frac{1}{a_{3}} (a′_{1}a′_{2}a′_{3})}σ̃_{105} ∈ 𝔗^{<105>}σ̃_{105} ⊂ Q_{3}.

Second, we wish to prove that given t_{1}, t_{2} ∈ {t_{−1−111}, t_{−11−11}, t_{1−1−11}}, the conjugations σ_{0}σ̃_{105}t_{1} and σ_{0}σ̃_{105}t_{2} are in the same orbit under the action of N(Q_{3}). We know that there is an index j ∈ {1, …, 1152} such that σ̃_{j}t_{1}σ̃_{j}^{−1} = t_{2} as in Subsection 5.1, but we wish also that σ̃_{j} commutes with σ̃_{105}. We find such indices by computer: j = 96 works for (t_{1}, t_{2}) = (t_{−1−111}, t_{−11−11}) and j = 122 works for (t_{−1−111}, t_{1−1−11}). From here, σ̃_{j}(σ_{0}σ̃_{105}t_{1})σ̃_{j}^{−1} = σ_{0}σ̃_{105}t_{2}, clearly being σ̃_{j} ∈ N(Q_{3}).

Third we state that for any t ∈ 𝔗^{<105>} ∩ 𝔗^{S^{1}} and for any u ∈ S^{1}, there exists μ ∈ N(Q_{3}) such that

(6)  μ(σ_{0}σ̃_{105}t)μ^{−1} = (σ_{0}σ̃_{105}t)t_{111u}.

Indeed, by taking a ∈ S^{1} such that āa^{−1} = u, the element μ := t_{111a} ∈ 𝔗^{<105>} ⊂ N(Q_{3}) verifies μ(σ_{0}σ̃_{105}t)μ^{−1} = σ_{0}σ̃_{105}tt_{111u}. From the last two points, the set ℭ_{Q_{3},1} \ σ_{0}(𝔗^{<105>} ∩ 𝔗^{S^{1}}) is contained in a same orbit under our action.

Finally, let us prove that there is one element in σ_{0}(𝔗^{<105>} ∩ 𝔗^{S^{1}}) connected to one in {σ_{0}σ̃_{105}t_{−11−1u}, σ_{0}σ̃_{105}t_{1−1−1u}, σ_{0}σ̃_{105}t_{−1−11u} : u ∈ S^{1}} by means of an element in N(Q_{3}).

Observe that the quasitorus generated by {t_{111u}, t_{−1−111}, σ̃_{105}} is a toral subquasitorus of Q_{3} as a consequence of [6, Lemma 2] and [7, Lemma 2], since it is isomorphic to ℂ^{×} × ℤ_{2}^{2}. Hence, [6, Proposition 7] gives us that there exists p ∈ aut_{ℂ}𝔣_{4} such that pQ_{3}p^{−1} ⊂ N(𝔗), p⟨t_{111u}, t_{−1−111}, σ̃_{105}⟩p^{−1} ⊂ 𝔗 and p commutes with t_{111u} and t_{−1−111}. So that pσ̃_{105}p^{−1} = t_{1} ∈ 𝔗 has order two and there are j ∈ {1, ..., 1152} and t_{2} ∈ 𝔗 with pt_{1−1−11}p^{−1} = σ̃_{j}t_{2} of order two. Thus pQ_{3}p^{−1} ⊂ Q(j, t_{2}). The nontorality implies that σ_{j} must belong to the orbit of σ_{k} for some index k ∈ {3, 15, 105, 106, 405}. Besides σ_{j} has order two, so k ∈ {105, 405}, and 𝔗^{<j>} contains some order not two elements (t_{111u}) so that k ≠ 405. As {t_{111u}, t_{−1−111}} commutes with σ̃_{j}, the only possibilities for j are {105, 144, 437, 468, 482, 514, 724, 748}, but among them only σ_{105} is conjugated to σ_{105}, forcing j = 105 and pQ_{3}p^{−1} = Q(105, t_{2}).

Now, by arguing as in [6, Proposition 6], we can find s ∈ 𝔗 such that t_{3} := (σ̃_{105}sσ̃_{105}s^{−1})t_{2} ∈ 𝔗^{<105>}. Let us check that μ := sp ∈ aut_{ℂ}𝔣_{4} is the required element. It verifies μt_{111u}μ^{−1} = t_{111u}, μt_{−1−111}μ^{−1} = t_{−1−111}, μσ̃_{105}μ^{−1} = t_{1} ∈ 𝔗^{<105>} and μt_{1−1−11}μ^{−1} = σ̃_{105}t_{3}, so that μQ_{3}μ^{−1} = Q_{3} and so μ ∈ N(Q_{3}). If μσ_{0}μ^{−1} = σ_{0}σ̃_{105}t_{4} with t_{4} ∈ 𝔗^{<105>} ∩ 𝔗^{S^{1}}, then σ_{0} belongs to the same orbit than σ_{0}σ̃_{105}t_{4} under the action and we have done. Otherwise, there is t_{4} ∈ 𝔗^{<105>} ∩ 𝔗^{S^{1}} such that μσ_{0}μ^{−1} = σ_{0}t_{4}. Then μ(σ_{0}σ̃_{105}t_{−1−111})μ^{−1} = (μσ_{0}μ^{−1})(μσ̃_{105}μ^{−1})(μt_{−1−111}μ^{−1}) = σ_{0}t_{4}t_{1}t_{−1−111} ∈ σ_{0}(𝔗^{<105>} ∩ 𝔗^{S^{1}}), obtaining again that there is just one orbit in ℭ_{Q_{3},1} and consequently that any grading on 𝔣_{4,4} inherited from a MAD-group conjugated to Q_{3} is isomorphic to the one inherited from Q_{3}.

Finally, let us consider some nonsplit and noncompact real form L which inherits the grading induced by Q_{3}. As ℭ_{Q_{3},3} = {σ_{0}σ̃_{105}t_{111u} : u ∈ S^{1}}, Equation (6) allows us to conclude as in the previous case that if L inherits another grading induced by some MAD-group conjugated to Q_{3}, the resulting grading is isomorphic to the original one.

5.4. Fine gradings inherited from the conjugation class of Q_{4}. Recall that Q_{4} = ⟨t_{1ϵϵ1}, t_{ϵ1ϵ^{2}ϵ^{2}}, σ̃_{15}⟩ with ϵ a primitive cubic root of the unit, and that any id ≠ f ∈ Q_{4} has order 3. Let us show that the fine grading on 𝔣_{4} induced by Q_{4} is not inherited by any of the real forms of 𝔣_{4}, that is, ℭ_{Q_{4}} = ∅. Suppose that there exists a conjugation σ of 𝔣_{4} satisfying that. Taking into account Proposition 2, σσ̃_{15} = σ̃_{15}^{2}σ, σt_{1ϵϵ1} = t_{1ϵϵ1}^{2}σ and σt_{ϵ1ϵ^{2}ϵ^{2}} = t_{ϵ1ϵ^{2}ϵ^{2}}^{2}σ. By Lemma 1, σ_{0}σ̃_{15} = σ̃_{15}σ_{0}, σ_{0}t_{1ϵϵ1} = t_{1ϵϵ1}^{2}σ_{0} and σ_{0}t_{ϵ1ϵ^{2}ϵ^{2}} = t_{ϵ1ϵ^{2}ϵ^{2}}^{2}σ_{0}, so that the automorphism h := σσ_{0} ∈ aut_{ℂ}𝔣_{4} satisfies hσ̃_{15} = σ̃_{15}^{2}h, ht_{1ϵϵ1} = t_{1ϵϵ1}h and ht_{ϵ1ϵ^{2}ϵ^{2}} = t_{ϵ1ϵ^{2}ϵ^{2}}h. From here, h^{2} commutes with any element in Q_{4} and then h^{2} ∈ Q_{4} (since Q_{4} is its own centralizer). As any element f ∈ Q_{4} satisfies f^{3} = id, we have h^{6} = id. Note that h^{3} ≠ id, since otherwise the facts h^{2}, id ∈ Q_{4} would imply h ∈ Q_{4} and then hσ̃_{15} = σ̃_{15}h. Thus the automorphism h^{3} ∈ Cent_{_{aut}_{ℂ}𝔣_{4}}(⟨t_{1ϵϵ1}, t_{ϵ1ϵ^{2}ϵ^{2}}⟩) has order just 2. Therefore the quasitorus ⟨h^{3}, t_{1ϵϵ1}, t_{ϵ1ϵ^{2}ϵ^{2}}⟩ is isomorphic to ℤ_{2} × ℤ_{3} × ℤ_{3} ≈ ℤ_{6} × ℤ_{3} and hence it is toral, because of [6, Lemma 2]. This is equivalent to saying that h^{3} belongs to the identity component of Cent_{_{aut}_{ℂ}𝔣_{4}}(⟨t_{1ϵϵ1}, t_{ϵ1ϵ^{2}ϵ^{2}}⟩) =: Z. We also have that (h^{3}σ̃_{15})^{2} = h^{6} = id and arguing as for h^{3} we get that h^{3}σ̃_{15} also belongs to the identity component of Z. Both observations imply that σ̃_{15} belongs to the identity component of Z too. But that means that ⟨σ̃_{15}, t_{1ϵϵ1}, t_{ϵ1ϵ^{2}ϵ^{2}}⟩ = Q_{4} is toral, of course false. Therefore, any of the real forms of 𝔣_{4} does not inherit the fine grading on 𝔣_{4} induced by any quasitorus conjugated to Q_{4}.

In the following theorems we compile the above information about the problem of the uniqueness and about which of the fine gradings on 𝔣_{4} are inherited by some of its real forms.

Theorem 1. Let L be a real form of 𝔣_{4} such that inherits the fine grading induced by a MAD-group Q of aut_{ℂ}𝔣_{4}. If L inherits the fine grading induced by a MAD-group conjugated to Q, then both gradings are isomorphic.

Theorem 2. The fine gradings on 𝔣_{4} =: S which are inherited by some of its real forms are precisely:

Q_{1} Q_{2} Q_{3} Q_{4}
ℤ^{4}-grading ℤ_{2}^{5}-grading ℤ_{2}^{3} × ℤ-grading ℤ_{3}^{3}-grading
𝔣_{4,4}  Yes  Yes  Yes  No
𝔣_{4,−52}  No  Yes  No  No
𝔣_{4,−20}  No  Yes  Yes^{∗}  No

∗: the nonsplit and noncompact real form which inherits Q_{3} is not the standard 𝔣_{4,−20} = S^{θ} but S^{σ_{0}σ̃_{105}}.

5.5. All the fine gradings on the real forms of 𝔣_{4} induce fine gradings on 𝔣_{4}. We have finished the first step in our program. The second and final step consists of studying if these real forms inherit some non-fine grading on 𝔣_{4} =: S which, however, gives rise to a fine grading on the real forms.

Lemma 2. Let σ be a conjugation of 𝔣_{4} and suppose that the automorphism f := σ_{0}σ of 𝔣_{4} is semisimple. Then S^{σ} inherits the grading induced by f on 𝔣_{4} if and only if Spec(f) ⊂ S^{1}.

Proof. Denote by V_{λ,f} the eigenspace of f associated to the eigenvalue λ, and by Spec(f) the set of eigenvalues of f. Note that, if v ∈ V_{λ,f}, then σ_{0}fσ_{0}(v) = f^{−1}(v) = \frac{1}{λ}v so that σ_{0}(V_{λ,f}) ⊂ V_{1/λ̄,f}. Now S^{σ} inherits the grading induced by f if and only if σ(V_{λ,f}) ⊂ V_{λ,f} for any λ ∈ Spec(f). As σ = σ_{0}f, this is equivalent to σ_{0}(V_{λ,f}) ⊂ V_{λ,f} for any λ ∈ Spec(f), and hence, to λ = 1/λ̄ for any λ ∈ Spec(f). □

Proposition 6. Let σ be a conjugation of 𝔣_{4} such that S^{σ} inherits the grading induced by a quasitorus Q of aut_{ℂ}𝔣_{4}. Then S^{σ} inherits the grading induced by a MAD-group Q̃ of aut_{ℂ}𝔣_{4} such that Q ⊂ Q̃.

Proof. As any MAD-group is conjugated to some Q_{i}, i = 1, 2, 3, 4, there is no loss of generality in supposing that σ inherits the grading induced by a quasitorus Q ⊂ Q_{i}, i = 1, 2, 3, 4. If Q = Q_{i} we take Q̃ = Q_{i} and the assertion is proved. Then, let us consider the case Q ⊊ Q_{i}. If i = 4, Q is toral by [6, Lemma 2] and so this case is reduced to those in which Q ⊊ Q_{1} = 𝔗. Hence, from now on i = 1, 2, 3.

Let κ be the Killing form of S = 𝔣_{4}. It is well known that the map < ·, · >: S × S → ℂ defined by < x, y >= −κ(x, τ(y)) is a definite positive hermitian form (see for instance [9, p. 92]), where τ denotes as always the standard compact conjugation. For any conjugation σ_{1} of S, the automorphism σ_{1}τ is selfadjoint with respect to < ·, · > ([9, Apendix A, 1.7 Theorem]). Thus, there is an orthonormal basis of S over ℂ, relative to < ·, · >, formed by eigenvectors for σ_{1}τ. The corresponding eigenvalues are all real and nonzero.

Let us show that we can suppose that σ and τ commute. Indeed, consider the automorphism f := σ_{0}σ of S. Since S^{σ_{0}} (see Theorem 2) and S^{σ} inherit the grading induced by Q, then Proposition 3 gives us f ∈ Cent(Q). Hence (στ)^{2} = fωfω ∈ Cent(Q) (since ωt = t^{−1}ω for any t ∈ 𝔗 and ω commutes with σ̃_{i}), being (στ)^{2} a semisimple automorphism of S with spectrum contained in ℝ_{>0} by the observations in the previous paragraph. Hence, there exists B a ℂ-basis of S in such a way that (στ)^{2} and Q are simultaneously diagonalizable relative to B. By arguing as in [9, Apendix A, Proof of 1.7 Theorem], if the matrix of (στ)^{2} relative to B is diag {α_{1}, …, α_{52}}, we can take φ ∈ aut_{ℂ}S whose matrix relative to B is the exponential of diag \{\frac{1}{4}log α_{1}, …, \frac{1}{4}log α_{52}\}, which satisfies that φ^{−1}σφ commutes with τ and with φ^{−1}Qφ = Q (φ and Q are simultaneously diagonalizable). From here, we can suppose στ = τσ, by replacing σ by φ^{−1}σφ and taking into account that φ^{−1}σφ inherits φ^{−1}Qφ = Q.

Let us prove then that Spec (f) ⊂ S^{1}. Since f = σ_{0}σ, we have f^{−1} = σσ_{0} and f^{−1}ω = σσ_{0}ω = στ = τσ = ωσ_{0}σ = ωf, so that

(7)  f^{−1} = ωfω.

As we can write τσ = ωf and, by the second paragraph in this proof, we also have that τσ is selfadjoint relative to < ·, · >, then we have ωf = (ωf)^{♯} = f^{♯}ω^{♯}, where f^{♯} denotes the adjoint of f relative to < ·, · >. But ω = τσ_{0} is also selfadjoint, hence f^{♯} = ωfω, which, jointly with Equation (7), yield f^{♯} = f^{−1}. Thus, f is semisimple and Spec (f) ⊂ S^{1}. Lemma 2 implies now that S^{σ} inherits the grading induced by Q′, where Q′ is the quasitorus generated by Q ∪ {f}. If f ∈ Q, we can apply Proposition 3 to see that S^{σ} inherits the grading induced by Q_{i}, since S^{σ_{0}} inherits the grading induced by Q_{i} (Theorem 2) and σ = σ_{0}f with f ∈ Q ⊂ Q_{i} = Cent(Q_{i}). In case that f ∉ Q (that is, Q ⊊ Q′), we can argue with Q′ in a similar way as we did with Q. This procedure has to finish because MAD-groups are finitely generated. □

As a consequence, any real form S^{σ} of 𝔣_{4} satisfies that there is not a non-fine grading on 𝔣_{4} which is inherited by S^{σ} and gives rise to a new fine grading on the real form S^{σ}. Hence, we have proved the following main result.

Theorem 3. The only fine gradings on the real forms of 𝔣_{4} are those described in Theorem 2.

Remark 3. We want to remark that we have got all the gradings on the real forms of the Albert algebra, taking into account the following facts: First, in Section 7 we provide realizations of all the gradings appearing in Theorem 2 (with label “Yes”) not only for the real forms of 𝔣_{4} but for the corresponding real forms of the Albert algebra. Second, there do not exist realizations of the gradings labeled with “No” in Theorem 2 neither in the corresponding real forms of the Albert algebra (a Ggrading on J would induce one in Der(J)). Third, there is only one fine grading for each label “Yes” in Theorem 2, because in the arguments of the proof we worked inside N(𝔗) and all the arguments are true replacing the torus of aut_{ℂ}𝔣_{4} by the torus of aut_{ℂ}J (in fact, a description of the images in aut_{ℂ}J of the elements σ̃_{i} can also be download from [23]). And fourth and finally, the proof of Proposition 6 does not use anything so specific to 𝔣_{4} that it can not be applied also to the Albert algebra. Therefore we have got theorems analogous to Theorem 1, Theorem 2 and Theorem 3 for the Albert algebra case.

In the next section, we prove again this result on the gradings on the real forms of the Albert algebra as a direct consequence of a more general approach to gradings. Looking at the gradings by means of affine algebraic group schemes helps to understand that this close relationship is not casual but it obeys a deep reason.

6. Squeme techniques. Fine gradings on the real forms of 𝔤_{2} and on the Albert algebra.

In this section we look at gradings on algebras from a different approach, which can be applied in different settings. As examples of applications we study the fine gradings on the real forms of 𝔤_{2} and of the Albert algebra.

6.1. Let k be an arbitrary field, and alg_{k} and grp the categories of (unital commutative associative) k-algebras and groups respectively. Let G: alg_{k} → grp be an affine group scheme and consider a G-module V or equivalently a representation ρ: G → GL(V) which is a homomorphism of group functors into GL(V): alg_{k} → grp where GL(V)(R) := GL(V_{R}) is the group of R-linear automorphisms of V_{R} := V ⊗_{k} R. As it is well known, the representation ρ induces a comodule map σ: V → V ⊗ k[G] where k[G] is the Hopf algebra representing G. Recall that σ = ρ_{k[G]}(1_{k[G]})i where i: V → V ⊗ k[G] is the canonical inclusion. Moreover σ satisfies

(8)  (σ ⊗ 1_{k[G]})σ = (1_{V} ⊗ Δ)σ, (1_{V} ˆ⊗ϵ)σ = 1_{V}

where Δ is the comultiplication of the Hopf algebra k[G] and ϵ its counit or augmentation. It is well known that all the information of the representation ρ can be recovered from the single comodule map σ (see [21, 2.8, p. 27]). So σ determines completely the representation and for some purpose we can forget ρ and work with the comodule map alone.

If A is an algebra over k, we can consider the group functor aut(A): alg_{k} → grp. This group functor maps any k-algebra R to the group aut_{R} A_{R} of R-automorphisms of A_{R} := A ⊗_{k} R. Given an abelian group Λ we can construct the diagonalizable affine group scheme Diag(Λ): alg_{k} → grp such that Diag(Λ)(R) := hom_{k}(k[Λ], R) where k[Λ] is the group algebra of Λ endowed with its usual Hopf algebra structure. Of course Diag(Λ)(R) can be identified with the group of all group homomorphisms φ: Λ → R^{×} with multiplication φφ′(λ) := φ(λ)φ′(λ) for all λ ∈ Λ.

If A = ⊕_{λ∈Λ}A_{λ} is a grading on A over the group Λ, then we can define a homomorphism of group functors ρ: Diag(Λ) → aut(A) such that for any φ ∈ Diag(Λ)(R), the automorphism ρ_{R}(φ): A_{R} → A_{R} is defined by ρ_{R}(φ)(a_{λ} ⊗ 1) := a_{λ} ⊗ φ(λ) for any a_{λ} ∈ A_{λ} and λ ∈ Λ. Thus A_{λ} agrees with the k-subspace of A formed by all elements a ∈ A such that ρ_{R}(φ)(a ⊗ 1) = a ⊗ φ(λ) for all φ ∈ Diag(Λ)(R). Reciprocally any homomorphism ρ: Diag(Λ) → aut(A) of group functors induce a grading on A so that A = ⊕_{λ∈Λ}A_{λ} and each A_{λ} is given as above. The group functor homomorphism ρ may be seen as a representation of Diag(Λ) on A. As previously mentioned, all the information of this representation is condensed in the comodule action σ: A → A ⊗ k[G] defined above. It is easy to see that in this case σ: A → A ⊗ k[Λ] is a k-algebra homomorphism satisfying (8) and σ(a_{λ}) = a_{λ} ⊗ λ for any a_{λ} ∈ A_{λ}. Conversely given any k-algebra homomorphism satisfying (8) we get a Λ-grading on A such that A_{λ} := {a ∈ A: σ(a) = a⊗λ} for all λ ∈ Λ. Summarizing the previous paragraphs it should be clear that independently of the nature of the ground field k and of the dimension of the k-algebra A, any Λ-grading induces a k-algebra homomorphism σ: A → A⊗k[Λ] satisfying (8) given by σ(a) = a ⊗ λ for a ∈ A_{λ} and λ ∈ Λ. Such a map is called a comodule map and what is even more important is that any Λ-grading arises in this way (see also [20, Proposition 4.8]).

One final comment to these remarks. If A and B are k-algebras and θ: aut(A) ≅ aut(B) is an isomorphism of k-group functors, then the fine gradings on A and B are in bijective correspondence.

6.2. Some correspondences of gradings. We now apply some of the above ideas. For this, we must recall first that the complex octonion algebra C (isomorphic to the Zorn matrix algebra), whose automorphism group is G_{2} = aut(C), and the Albert algebra J = H_{3}(C), whose automorphism group is F_{4}, provide isomorphisms of linear algebraic groups

(9)  G_{2} ≅ aut𝔤_{2},  F_{4} ≅ aut𝔣_{4}

where 𝔤_{2} = Der(C) and 𝔣_{4} = Der(J).

The following result provides a useful criterium for isomorphisms of algebraic groups schemes. It will be of interest for our purposes:

Proposition 7. ([22, (22.5) Proposition, p. 340]) Let f: G → H a homomorphism of k-algebraic group schemes with H smooth. Then the following conditions are equivalent:

(1) f is an isomorphism.

(2) f is injective and surjective.

(3) falg : G(kalg) → H(kalg) is an isomorphism and df is injective

In the above proposition the notation kalg means the algebraic closure of k.

Theorem 4. Let V = C (complex octonion algebra) or V = J = H_{3}(C) (complex Albert algebra). Let U be any real form of V and D = Der(U) its Lie algebra of derivations. There is a bijective correspondence from the equivalence classes of fine gradings on U to that on D coming from an isomorphism of algebraic group schemes aut(U) ≅ aut(D).

Proof. Consider the homomorphism of ℝ-algebraic group schemes

Ad: aut(U) → aut(D)

such that for any real algebra R and ϕ ∈ aut U_{R}, we have Ad_{R}(ϕ): D_{R} → D_{R}, given by d ↦ ϕdϕ^{−1} (the standard identification of Der(U_{R}) with Der(U)_{R} has been used). In this case, since U_{ℂ} is the complex algebra C (respectively J) and D_{ℂ} = 𝔤_{2} (respectively D_{ℂ} = 𝔣_{4}), the homomorphism Ad_{ℂ}: aut U_{ℂ} → aut D_{ℂ} is some of the isomorphisms (9). Furthermore, the Lie algebra of aut U is a form of 𝔤_{2} (respectively 𝔣_{4}) while the Lie algebra of aut D is a form of Der(𝔤_{2}) (or of Der(𝔣_{4})) and the differential dAd = ad is an isomorphism. Thus the conditions in (3) of Proposition 7 are satisfied, which implies that Ad: aut(U) ≅ aut(D) is an isomorphism of algebraic group schemes. □

Let us apply that, as a corollary of the previous theorem, there is a bijective correspondence of equivalence classes of fine gradings between the fine gradings on U and those on D. As it is well known the real forms of the complex octonions algebra C (respectively the complex Albert algebra) are in bijective correspondence with the real forms of the complex algebra 𝔤_{2} (resp. the real forms of 𝔣_{4}). There are two real forms 𝕆 and 𝕆_{s} of the complex octonion algebra C. The first of them is 𝕆 the division algebra of real octonions while the second is 𝕆_{s} the algebra of real split octonions isomorphic to the real Zorn matrices algebra. Elduque computed the gradings on all octonion algebras and as a consequence of his classification (see [8]) we know that up to equivalence: (1) there is only a fine grading on 𝕆 which is the ℤ_{2}^{3}-grading given in [8, Theorem 8, p. 350], and (2) there are two fine gradings on 𝕆_{s}. One is the ℤ^{2}-grading given in [8, Theorem 4, p. 346], in which the zero component is the plane generated by the orthogonal idempotents e_{1} and e_{2}, while the rest of the standard basis elements generate one-dimensional homogeneous components. The other one is again the ℤ_{2}^{3}-grading described in [8, Theorem 8, p. 350]. Therefore

Theorem 5. Let L be a real form of 𝔤_{2}. Then

(1) If L is split, any fine grading on L is, up to equivalence, isomorphic either to the ℤ^{2}-grading given by any Cartan decomposition of L, or to the ℤ_{2}^{3}- grading induced by the one on 𝕆_{s}.

(2) If L is compact, any fine grading on L is, up to equivalence, isomorphic to the ℤ_{2}^{3}-grading induced by the one on 𝕆.

Remark 4. If we want to understand this theorem in the same terms as for 𝔣_{4}, consider B = {e_{1}, e_{2}, u_{1}, u_{2}, u_{3}, v_{1}, v_{2}, v_{3}} ([8]) the so called standard basis of C the algebra of complex octonions. Recall that the derivations D_{x,y} = [l_{x}, l_{y}] + [l_{x}, r_{y}] + [r_{x}, r_{y}] generate 𝔤_{2} = Der(C) if x, y ∈ C, where r_{x} and l_{x} denote the right and left multiplication operators respectively, and that 𝔥_{𝔤_{2}} = ∑_{1}^{3} ℂD_{u_{i},v_{i}} is a Cartan subalgebra of L = 𝔤_{2}. Take σ_{0}, ω and τ as in Subsection 2.3 for a choice of canonical generators of 𝔤_{2} respect to a set of simple roots relative to 𝔥_{𝔤_{2}}. Take T the maximal torus of G_{2} of all automorphisms acting as the identity on 𝔥_{𝔤_{2}}. If t_{x,y} is the automorphism whose matrix relative to B is the diagonal matrix

diag(1, 1, x, y, (xy)^{−1}, x^{−1}, y^{−1}, xy),

then T = {t_{x,y} : x, y ∈ ℂ^{×}}. According to [5], the MAD-groups of aut_{ℂ}𝔤_{2} are given by

\begin{matrix}P_{1} = T ≅ (ℂ^{×})^{2}, \\ P_{2} = \{t_{−1,1}, t_{1,−1}, ω\} ≅ ℤ_{2}^{3}, \end{matrix}

and, as in Lemma 1,

\begin{matrix}σ_{0}t_{x,y} = t_{\overline{x}, \overline{y}}σ_{0}, \\ ωt_{x,y} = t_{x^{−1}, y^{−1}}ω, \end{matrix}

so that σ_{0} inherits P_{1} and P_{2} and the compact conjugation τ = σ_{0}ω inherits only P_{2}. Then it is quite easy to prove directly the following theorem, equivalent to Theorem 5, as well as the uniqueness of the inherited gradings from a conjugacy class:

Theorem 6. The fine gradings on 𝔤_{2} which are inherited by some of its real forms are precisely:

P_{1}  P_{2}
ℤ^{2}-grading ℤ_{2}^{3}-grading
S^{σ_{0}}  Yes  Yes
S^{τ}  No  Yes

With this remark, we have wanted to emphasize that it is possible and easy to apply our techniques on Sections 2 and 3 to the case of 𝔤_{2}, although using the relationship between gradings and squemes is a more conceptual approach. The same fact happens when considering real forms of the Albert algebra, as mentioned in Remark 3: we can get the gradings directly with the previous techniques but there is nothing to do if we apply the squemes point of view:

As we have found all the equivalence classes of fine gradings on the real forms of 𝔣_{4}, we can describe the fine gradings on the real forms of the complex Albert algebra J, because Theorem 4 gives us a way to translate fine gradings on a real form J_{ℝ} of J to Der(J_{ℝ}) (a real form of 𝔣_{4}) and conversely. In this context, we will say that J_{ℝ} has the fine grading induced by the one on Der(J_{ℝ}). We will also say that the real form J_{ℝ} of the complex Albert algebra J is split (respectively compact, nonsplit and noncompact) if its derivation algebra is isomorphic to a split real form of 𝔣_{4} (respectively compact, nonsplit and noncompact).

Theorem 7. Any fine grading on a real form J_{ℝ} of the complex Albert algebra J is, up to equivalence, isomorphic to

1. If J_{ℝ} is split, either the ℤ^{4}-grading, or the ℤ_{2}^{5}-grading or the ℤ_{2}^{3} ×ℤ-grading induced by the ones of 𝔣_{4,4} given in Theorem 2.

2. If J_{ℝ} is compact, the ℤ_{2}^{5}-grading induced by the one of 𝔣_{4,−52} given in Theorem 2.

3. If J_{ℝ} is nonsplit and noncompact, either the ℤ_{2}^{5}-grading or the ℤ_{2}^{3} × ℤgrading induced by the ones of 𝔣_{4,−20} given in Theorem 2.

7. Models of the fine gradings on the real forms of 𝔣_{4} and of the Albert algebra

In this section we give a detailed description of the fine gradings on the real forms of 𝔣_{4} which have been obtained in Theorem 2, as well as on the real forms of the Albert algebra described in Theorem 7. All such gradings can be described without using the automorphisms in N(𝔗) producing the gradings. Instead of that, we make use of the construction of the exceptional Lie algebra 𝔣_{4} and its real forms in terms of symmetric composition algebras given in [10] and [11] and a modified version of the construction of the Albert algebra given in [12].

7.1. Models of the real forms by means of symmetric composition algebras. We will follow the notation [10, §2] for the basic definitions and results on symmetric composition algebras.

Given a real symmetric composition algebra (S, ∗, q) of dimension 8, we can slightly modify the construction of the Jordan algebra 𝔸 given in [12, §2], by a set of three nonzero scalars ε = (ε_{0}, ε_{1}, ε_{2}) ∈ {±1}^{3}. Thus, we define the commutative algebra

𝔸_{(ε_{0},ε_{1},ε_{2})}(S) := ℝ^{3} ⊕ (⨁_{i=0}^{2} ι_{i}(S)),

where ι_{i}(S) is just a copy of S (i=0,1,2) and the product is given by

• (α_{0}, α_{1}, α_{2})(β_{0}, β_{1}, β_{2}) = (α_{0}β_{0}, α_{1}β_{1}, α_{2}β_{2}),
• (α_{0}, α_{1}, α_{2})ι_{i}(x) = \frac{1}{2}(α_{i+1} + α_{i+2})ι_{i}(x),
• ι_{i}(x)ι_{i+1}(y) = ε_{i+2}ι_{i+2}(x ∗ y),
• ι_{i}(x)ι_{i}(y) = 2 ε_{i+1}ε_{i+2}q(x, y)(e_{i+1} + e_{i+2}),

where the indices have been taken modulo 3 and {e_{0}, e_{1}, e_{2}} is the canonical basis of ℝ^{3}. This algebra 𝔸_{ε}(S) is a real form of the Albert algebra.

Now consider the orthogonal Lie algebra 𝔬(S, q) = {d ∈ End_{ℝ}(S) : q(d(x), y) + q(x, d(y)) = 0 ∀x, y ∈ S}, and the subalgebra of 𝔬(S, q)^{3} defined by

𝔱𝔯𝔦(S, ∗, q) = {(d_{0}, d_{1}, d_{2}) ∈ 𝔬(S, q)^{3} : d_{0}(x ∗ y) = d_{1}(x) ∗ y + x ∗ d_{2}(y) ∀x, y ∈ S},

called the triality algebra. One can form (see [10, §3]) the ℤ_{2}^{2}-graded Lie algebra 𝔤_{ε}(S, ℝ) defined as

𝔤_{ε}(S, ℝ) = 𝔱𝔯𝔦(S, ∗, q) ⊕ ι_{0}(S) ⊕ ι_{1}(S) ⊕ ι_{2}(S),

where the bracket is given by • 𝔱𝔯𝔦(S, ∗, q) is a Lie subalgebra of 𝔤(S, ℝ),

• [(d_{0}, d_{1}, d_{2}), ι_{i}(x)] = ι_{i}(d_{i}(x)),
• [ι_{i}(x), ι_{i+1}(y)] = ε_{i+2}ι_{i+2}(x ∗ y),
• [ι_{i}(x), ι_{i}(y)] = ε_{i+1}ε_{i+2}θ^{i}(t_{x,y}),

where t_{x,y} is the element of 𝔱𝔯𝔦(S, ∗, q) defined by

t_{x,y} = (σ_{x,y}, \frac{1}{2}q(x, y)id − r_{x}l_{y}, \frac{1}{2}q(x, y)id − l_{x}r_{y}),

with σ_{x,y}(z) = q(x, z)y−q(y, z)x, r_{x}(z) = z∗x and l_{x}(z) = x∗z; and where θ denotes the order 3 automorphism of 𝔱𝔯𝔦(S, ∗, q) defined by θ(d_{0}, d_{1}, d_{2}) := (d_{2}, d_{0}, d_{1}).

It is well known that 𝔤_{ε}(S, ℝ) is a real form of 𝔣_{4}. More precisely, if p𝕆 and p𝕆_{s} denote the real eight-dimensional symmetric composition algebras of the real split paraoctonions and the real division paraoctonions respectively, then, according to [11, §5],

\begin{matrix}𝔤_{ε}(S, ℝ) & S = p𝕆_{s} & S = p𝕆 \\ ε = (1, 1, 1) & 𝔣_{4,4} & 𝔣_{4,−52} \\ ε = (1, −1, −1) & 𝔣_{4,4} & 𝔣_{4,−20} \end{matrix}

Moreover, note that both constructions of the real forms of the Albert algebra and of 𝔣_{4} are directly related, because there is a Lie algebra isomorphism between Der(𝔸_{ε}(S)) and 𝔤_{ε}(S, ℝ). More precisely, the isomorphism is given by the map

𝔤_{ε}(S, ℝ) → Der(𝔸_{ε}(S))

defined by

(d_{0}, d_{1}, d_{2}) ∈ 𝔱𝔯𝔦(S, ∗, q) ↦ D_{(d_{0},d_{1},d_{2})}

where D_{(d_{0},d_{1},d_{2})} is the derivation of 𝔸_{ε}(S) given by D_{(d_{0},d_{1},d_{2})}(α_{0}, α_{1}, α_{2}) = 0 and D_{(d_{0},d_{1},d_{2})}(ι_{i}(x)) = ι_{i}(d_{i}(x)), and

ι_{i}(x) ↦ D_{i}(x)

where D_{i}(x) is the derivation of 𝔸_{ε}(S) given by D_{i}(x) = 2[l_{ι_{i}(x)}, l_{e_{i+1}}], for l the operator left multiplication. In particular we have realizations of the three types of real forms of the Albert algebra.

7.2. Realizations of the ℤ_{2}^{5}-gradings. Independently of the considered symmetric algebra S and also of ε, we have a natural ℤ_{2}^{2}-grading in the constructions in the previous subsection, both in 𝔸_{ε}(S) as in 𝔤_{ε}(S, ℝ). They are given by

\begin{matrix}𝔤(S, ℝ)_{(0,0)} = 𝔱𝔯𝔦(S, ∗, q), & 𝔸_{ε}(S)_{(0,0)} = ℝ^{3}, \\ 𝔤(S, ℝ)_{(0,1)} = ι_{0}(S), & 𝔸_{ε}(S)_{(0,1)} = ι_{0}(S), \\ 𝔤(S, ℝ)_{(1,0)} = ι_{1}(S), & 𝔸_{ε}(S)_{(1,0)} = ι_{1}(S), \\ 𝔤(S, ℝ)_{(1,1)} = ι_{2}(S), & 𝔸_{ε}(S)_{(1,1)} = ι_{2}(S). \end{matrix}

Now we can mix both ℤ_{2}^{2}-gradings with the ℤ_{2}^{3}-fine gradings on p𝕆 and p𝕆_{s} (see [13] for the gradings on symmetric composition algebras), getting ℤ_{2}^{5}-gradings on 𝔸_{ε}(S) and on Der(𝔸_{ε}(S)). In particular we have a description of the ℤ_{2}^{5}-gradings on Der(𝔸_{(1,1,1)}(p𝕆_{s})) = 𝔣_{4,4}, Der(𝔸_{(1,1,1)}(p𝕆)) = 𝔣_{4,−52} and Der(𝔸_{(−1,1,−1)}(p𝕆)) = 𝔣_{4,−20} and on the three real forms of the Albert algebra.

7.3. Realizations of the ℤ_{2}^{3} × ℤ-gradings. If S is a para-Hurwitz algebra with standard involution −, and ε = (−1, 1, −1), the eigenvalues of the derivation D_{1}(1) are all integers numbers, hence the diagonalization of 𝔸_{ε}(S) relative to D_{1}(1) is a ℤ-grading 𝔸_{ε}(S) = 𝔸_{ε}(S)_{−2} ⊕ 𝔸_{ε}(S)_{−1} ⊕ 𝔸_{ε}(S)_{0} ⊕ 𝔸_{ε}(S)_{1} ⊕ 𝔸_{ε}(S)_{2}, given by

𝔸_{ε}(S)_{−2} = ⟨2(e_{0} − e_{2}) − ι_{1}(1)⟩,
𝔸_{ε}(S)_{−1} = ⟨ι_{0}(x) − ι_{2}(\overline{x}) : x ∈ S⟩,
𝔸_{ε}(S)_{0} = ⟨e_{1}, e_{0} + e_{2}, ι_{1}(x) : x ∈ S_{0}⟩,
𝔸_{ε}(S)_{1} = ⟨ι_{0}(x) + ι_{2}(\overline{x}) : x ∈ S⟩,
𝔸_{ε}(S)_{2} = ⟨2(e_{0} + e_{2}) + ι_{1}(1)⟩,

where S_{0} = \{x ∈ S : \overline{x} = −x\}.

Now this ℤ-grading induces a ℤ-grading on Der(𝔸_{ε}(S)), or equivalently, a ℤgrading on 𝔤_{ε}(S, ℝ), just the diagonalization relative to [ι_{1}(1), ·] ∈ Der(𝔤_{ε}(S, ℝ)). Both ℤ-gradings are compatible with the ℤ_{2}^{3}-gradings on p𝕆 and p𝕆_{s}, thus getting ℤ_{2}^{3} × ℤ-gradings on Der(𝔸_{(−1,1,−1)}(p𝕆_{s})) = 𝔣_{4,4} and Der(𝔸_{(−1,1,−1)}(p𝕆)) = 𝔣_{4,−20}, and on the corresponding two real forms of the Albert algebra.

Summarizing, if S = ⨁_{g∈ℤ_{2}^{3}} S_{g} is the ℤ_{2}^{3}-grading on S ∈ {p𝕆, p𝕆_{s}} described in [13] and 1 is the paraunit of S, we can assert:

Corollary 1. Any fine grading on a real form J of the complex Albert algebra J is isomorphic to one of the following:

1. If J is split,

(i) The ℤ^{4}-grading corresponding to the Cartan decomposition (see [6, p. 868]).

(ii) The ℤ_{2}^{5}-grading on J := 𝔸_{(1,1,1)}(p𝕆_{s}):

J_{(0,0,e)} = ℝ^{3},
J_{(0,0,g)} = 0, (g ≠ e)
J_{(0,1,g)} = ι_{0}((p𝕆_{s})_{g}),
J_{(1,0,g)} = ι_{1}((p𝕆_{s})_{g}),
J_{(1,1,g)} = ι_{2}((p𝕆_{s})_{g}).

(iii) The ℤ × ℤ_{2}^{3}-grading on J := 𝔸_{(−1,1,−1)}(p𝕆_{s}):

J_{(−2,e)} = ⟨2(e_{0} − e_{2}) − ι_{1}(1)⟩,
J_{(−2,g)} = 0, (g ≠ e)
J_{(−1,g)} = ⟨ι_{0}(x) − ι_{2}(\overline{x}) : x ∈ (p𝕆_{s})_{g}⟩,
J_{(0,e)} = ⟨e_{1}, e_{0} + e_{2}⟩,
J_{(0,g)} = ⟨ι_{1}(x) : x ∈ (p𝕆_{s})_{g}⟩, (g ≠ e)
J_{(1,g)} = ⟨ι_{0}(x) + ι_{2}(\overline{x}) : x ∈ (p𝕆_{s})_{g}⟩,
J_{(2,e)} = ⟨2(e_{0} − e_{2}) + ι_{1}(1)⟩,
J_{(2,g)} = 0, (g ≠ e).

2. If J is compact, (i) The ℤ_{2}^{5}-grading on J := 𝔸_{(1,1,1)}(p𝕆):

J_{(0,0,e)} = ℝ^{3},
J_{(0,0,g)} = 0, (g ≠ e)
J_{(0,1,g)} = ι_{0}((p𝕆)_{g}),
J_{(1,0,g)} = ι_{1}((p𝕆)_{g}),
J_{(1,1,g)} = ι_{2}((p𝕆)_{g}).

3. If J is nonsplit and noncompact, (i) The ℤ_{2}^{5}-grading on J := 𝔸_{(−1,1,−1)}(p𝕆):

J_{(0,0,e)} = ℝ^{3},
J_{(0,0,g)} = 0, (g ≠ e)
J_{(0,1,g)} = ι_{0}((p𝕆)_{g}),
J_{(1,0,g)} = ι_{1}((p𝕆)_{g}),
J_{(1,1,g)} = ι_{2}((p𝕆)_{g}).

(ii) The ℤ_{2}^{3} × ℤ-grading on J := 𝔸_{(−1,1,−1)}(p𝕆):

J_{(−2,e)} = ⟨2(e_{0} − e_{2}) − ι_{1}(1)⟩,
J_{(−2,g)} = 0, (g ≠ e)
J_{(−1,g)} = ⟨ι_{0}(x) − ι_{2}(\overline{x}) : x ∈ (p𝕆)_{g}⟩,
J_{(0,e)} = ⟨e_{1}, e_{0} + e_{2}⟩,
J_{(0,g)} = ⟨ι_{1}(x) : x ∈ (p𝕆)_{g}⟩, (g ≠ e)
J_{(1,g)} = ⟨ι_{0}(x) + ι_{2}(\overline{x}) : x ∈ (p𝕆)_{g}⟩,
J_{(2,e)} = ⟨2(e_{0} − e_{2}) + ι_{1}(1)⟩,
J_{(2,g)} = 0, (g ≠ e)

And in the same way we can have a detailed description of the fine gradings on the real forms of 𝔣_{4}.

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A. J. Calderón Martín: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, 11510 Puerto Real, Cádiz, Spain.

C. Draper Fontanals: Departamento de Matemática Aplicada, Escuela de las Ingenierías, Ampliación del Campus de Teatinos, Universidad de Málaga, 29071 Málaga, Spain.

C. Martín González: Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Campus de Teatinos, Universidad de Málaga, 29080 Málaga, Spain.