The étale Tits process of Jordan algebras revisited

Holger P. Petersson Fachbereich Mathematik FernUniversität - Gesamthochschule in Hagen D-58084 Hagen Germany e-mail: Holger.Petersson@FernUni-Hagen.de and Maneesh Thakur Harish-Chandra Research Institute Chhatnag Road, Jhusi Allahabad - 211019 India e-mail: maneesh@mri.ernet.in

Kurt Meyberg zum 65. Geburtstag gewidmet

0. Introduction

The étale Tits process, which was called the toral Tits process by Petersson-Racine [12], may be viewed as a Jordan-theoretical method to construct associative algebras with involution. More specifically, starting from a cubic étale algebra E and a quadratic étale algebra L, both over an arbitrary base field, as well as from invertible elements u ∈ E, b ∈ L having the same norms, the étale Tits process produces an absolutely simple Jordan algebra J = J(E, L, u, b) of degree 3 and dimension 9, which, by structure theory, must be the symmetric elements of a central simple associative algebra of degree 3 with involution of the second kind. In addition, E identifies canonically with a subalgebra of J.

Our first objective in the present paper is to prove the converse of this result: Given a cubic étale algebra E and a central simple associative algebra (B, τ) of degree 3 with involution of the second kind, every isomorphic embedding ι from E to J = H(B, τ), the Jordan algebra of τ-symmetric elements in B, will be shown in the extension theorem below to allow data L, u, b as above such that ι extends to an isomorphism from the étale Tits process J(E, L, u, b) onto J. Over fields of characteristic different from two and three, this result is due to Knus, Merkurjev, Rost and Tignol [4, (39.5), (6)]. Their proof rests on [4, (36.38)] and hence on certain generic choices which seem to require an ample supply of invertible elements, this being automatic only if the base field is big enough and hence, in particular, if J is a division algebra. Under the very last restriction, the extension theorem also derives quite easily from the classification theory of the Tits process [14, Theorem 3.1 (ii)]. On the other hand, as we shall see below, a unified proof working in full generality turns out to be surprisingly delicate.

The idea is to mimic McCrimmon’s classical approach [7] to the enumeration of Albert algebras over arbitrary base fields.

The extension theorem will be applied in two different ways. The first application (3.2) is concerned with cubic étale algebras E, E′ giving rise to étale Tits processes J, J′, respectively, and spells out necessary and sufficient conditions for an isomorphism E′ ⟶^{∼} E to be extendable to an isomorphism J′ ⟶^{∼} J. The second application (4.2) is concerned with attaching invariants to (isomorphic) embeddings from E to J = H(B, τ), where E is a cubic étale algebra and (B, τ) is a central simple associative algebra of degree 3 with involution of the second kind. These invariants, called norm classes, are analogues of the invariants that under the same name were attached by Albert-Jacobson [2, §9] to “ordered basic sets of idempotents in exceptional simple Jordan algebras”. In analogy to [2, Theorem 9] we show that two embeddings from E to J have the same norm class if and only if they are equivalent, i.e., can be transformed into one another by an automorphism of J. Thus the presence of distinct norm classes my be viewed as an obstruction to the validity of the Skolem-Noether Theorem.

1. The étale Tits process and the extension theorem

Throughout this paper we fix an arbitrary base field k. All algebras considered in the sequel (as well as subalgebras and homomorphisms thereof) are supposed to be unital. We write A^{×} for the set of invertible elements in an algebra A whenever this makes sense. The bilinearization of a quadratic map Q will be denoted by Q(x, y) = Q(x + y) − Q(x) − Q(y).

1.1 Cubic norm structures. Following McCrimmon [6] and Petersson-Racine [13], a cubic norm structure over k is a quadruple (V, N, ♯, 1) consisting of a vector space V over k, a cubic form N : V ⟶ k (the norm), a quadratic map ♯ : V ⟶ V, x ⟼ x^{♯}, (the adjoint) and a distinguished element 1 ∈ V (the base point) such that the relations

(1)  x^{♯♯} = N(x)x  (“adjoint identity”),

N(1) = 1, T (x^{♯}, y) = (DN)(x)y, 1^{♯} = 1, 1 × y = T (y)1 − y hold under all scalar extensions, where T = −(D^{2} log N)(1): V × V ⟶ k is the associated trace form, x × y = (x + y)^{♯} − x^{♯} − y^{♯} is the bilinearization of the adjoint and T (y) = T (1, y). Then the U-operator U_{x}y = T (x, y)x − x^{♯} × y and the base point 1 give V the structure of a unital quadratic Jordan algebra denoted by J(V, N, ♯, 1). From [6, (15), (21) and p. 501] we recall the equations

T (x × y, z) = T (x, y × z), (2)

x^{♯} = x^{2} − T (x)x + T (x^{♯})1, (3)

N(x × y) + N(x)N(y) = T (x^{♯}, y)T (x, y^{♯}). (4)

From [6, Theorem 2] we know that x ∈ J is invertible if and only if N(x) ≠ 0, in which case

x^{−1} = N(x)^{−1}x^{♯}. (5)

1.2 Étale algebras. Quadratic étale k-algebras are classified by H^{1}(k, ℤ/2ℤ). The element of H^{1}(k, ℤ/2ℤ) corresponding to a quadratic étale algebra L over k will be denoted by δ(L/k); it is basically the ordinary discriminant if k has characteristic not two. Conversely, we write k{δ} for the quadratic étale k-algebra corresponding to δ ∈ H^{1}(k, ℤ/2ℤ).

Let E be a cubic étale k-algebra. Then E carries a natural Jordan algebra structure (with U-operator U_{x}y = x^{2}y) which, in fact, agrees with J(V, N, ♯, 1) as in 1.1, where V is the underlying vector space, N is the norm, ♯ is the adjoint (i.e., the numerator of the inversion map), and 1 is the unit of E. By abuse of notation, we do not distinguish carefully between E as a cubic étale k-algebra and its induced Jordan algebra structure. The element of H^{1}(k, ℤ/2ℤ) corresponding to the discriminant algebra of E [4, §18] will be denoted by δ(E/k).

1.3 The étale Tits process. Referring to [13] for details, we briefly recall the main ingredients of the étale Tits process, called the toral Tits process in [12]. Let L, E be étale k-algebras of dimension 2,3, respectively. We write N_{E}, ♯, T_{E} both for the norm, adjoint, trace, respectively, of E and for their natural extensions to the cubic étale L-algebra E ⊗ L, unadorned tensor products always being taken over k. Analogous conventions apply to the norm N_{L}, the trace T_{L} and the nontrivial k-automorphism σ of L, E ⊗ L this time being viewed as a quadratic étale E-algebra. Notice that E identifies naturally as E ⊗ 1 with H(E ⊗ L, σ), the fixed points of E ⊗ L under the involution σ.

Now let u ∈ E, b ∈ L be invertible elements having the same norms, so N_{E}(u) = N_{L}(b) = bσ(b) ≠ 0. Extending N_{E}, ♯, 1 as given on E/k and E ⊗ L/L to the vector space V = E × (E ⊗ L) over k according to the rules

(6)  N((v, x)) = N_{E}(v) + bN_{E}(x) + σ(b)σ(N_{E}(x)) − T_{E}(v, xuσ(x)),
(7)  (v, x)^{♯} = (v^{♯} − xuσ(x), σ(b)σ(x)^{♯}u^{−1} − vx),
(8)  1 = (1, 0)

for v ∈ E, x ∈ E ⊗ L, we obtain a cubic norm structure (V, N, ♯, 1) as in 1.1 whose associated trace form is given by

T ((v, x), (w, y)) = T_{E}(v, w) + T_{E}(xu, σ(y)) + T_{E}(yu, σ(x)) (9)

for v, w ∈ E, x, y ∈ E ⊗ L. The ensuing Jordan algebra will be denoted by J = J(E, L, u, b) = J(V, N, ♯, 1); it is said to arise from E, L, u, b by the étale Tits process. Following [14, Corollary 4.3], J is an absolutely simple Jordan algebra of degree 3 and dimension 9, hence by structure theory (cf., e.g., [8, (15.7)] and [16, Theorem 1]) must be the symmetric elements of a central simple associative algebra of degree 3 with involution of the second kind. In particular, N = N_{J} is the generic norm and T = T_{J} is the generic trace of J. Clearly, E identifies with a subalgebra of J through the first factor. The relation

v × (0, x) = (0, −vx) (10)

follows immediately by linearizing (7). Since the trace form of J is nondegenerate (by (9)), we deduce from [6, p. 507] that a linear map between étale Tits processes preserving norms and units is necessarily an isomorphism. By [13, Theorem 5.2], J is a division algebra (i.e., all nonzero elements in J are invertible) if and only if b ∉ N_{E}((E ⊗ L)^{×}).

The following result has been proved in [12]; see also [4, (39.5) (5)] for char k ≠ 2, 3.

1.4 Theorem. ([12, Theorem 1]) Let L, E be étale k-algebras of dimension 2, 3, respectively, and suppose u ∈ E, b ∈ L are invertible elements satisfying N_{E}(u) = N_{L}(b). If (B, τ) is a central simple associative algebra of degree 3 over k with involution of the second kind such that the étale Tits process J(E, L, u, b) becomes isomorphic to the Jordan algebra over k of τ-symmetric elements in B, then the centre of B as a quadratic étale k-algebras corresponds to the element

δ(E/k) + δ(L/k) ∈ H^{1}(k, ℤ/2ℤ).  □

1.5 The étale first Tits construction. Let E be a cubic étale k-algebra and α ∈ k^{×}. Following [13, Theorem 3.5], we may form the étale first Tits construction J = J(E, α) = J(V, N, ♯, 1), where N_{E}, ♯, 1 as given on E extend to the vector space V = E × E × E over k according to the rules

(11)  N((v_{0}, v_{1}, v_{2})) = N_{E}(v_{0}) + αN_{E}(v_{1}) + α^{−1}N_{E}(v_{2}) − T_{E}(v_{0}v_{1}v_{2}),
(12)  (v_{0}, v_{1}, v_{2})^{♯} = (v_{0}^{♯} − v_{1}v_{2}, α^{−1}v_{2}^{♯} − v_{0}v_{1}, αv_{1}^{♯} − v_{2}v_{0}),
1 = (1, 0, 0)

for v_{0}, v_{1}, v_{2} ∈ E. The associated trace form of J is given by

T (x, y) = T_{E}(v_{0}, w_{0}) + T_{E}(v_{1}, w_{2}) + T_{E}(v_{2}, w_{1}) (13)

for x = (v_{0}, v_{1}, v_{2}), y = (w_{0}, w_{1}, w_{2}) ∈ J. Clearly, E identifies with a subalgebra of J(E, α) through the zeroeth factor. The relations

v_{0} × (0, v_{1}, 0) = (0, −v_{0}v_{1}, 0), v_{0} × (0, 0, v_{2}) = (0, 0, −v_{2}v_{0}) (14)

are well known and follow easily from linearizing (12). Finally, we conclude from [13, Theorem 3.5 and Proposition 3.8^{1}] that there are natural identifications of étale Tits processes J(E, L, u, b) as in 1.3, with L ≅ k × k split, and étale first Tits constructions J(E, α) as above which respect the identifications of E in J(E, L, u, b), J(E, α), respectively. More specifically we have

J(E, α) = J(E, k × k, (1, 1), (α, α^{−1})).

In particular, J(E, α) is a division algebra if and only if α ∉ N_{E}(E^{×}).

We are now ready to state the first main result of the paper.

1In the proof there is a misprint: μ^{−1} should be replaced by u^{−1}.

1.6 Extension Theorem. Let E be a cubic étale k-algebra, (B, τ) a central simple associative algebra of degree 3 with involution of the second kind over k and suppose ι is an isomorphic embedding from E to J = H(B, τ), the Jordan algebra over k of τ-symmetric elements in B. Writing K for the centre of B and L for the quadratic étale k-algebra corresponding to the element

δ(K/k) + δ(E/k) ∈ H^{1}(k, ℤ/2ℤ), (15)

there are invertible elements u ∈ E, b ∈ L satisfying N_{E}(u) = N_{L}(b) such that ι extends to an isomorphism from the étale Tits process J(E, L, u, b) onto J.

1.7 Remarks. The proof of the extension theorem will occupy the next section. The specific choice of L is forced upon us by 1.4. We may clearly assume that E ⊂ J is a subalgebra and ι is the inclusion.

1.8 Special cases. a) If k is an infinite (or a sufficiently big finite) field of characteristic not 2 or 3, the extension theorem follows from [4, (39.5), (5), (6) and (36.38)].

b) More specially, if J is a division algebra, the extension theorem is also implied by the general theory of the Tits process. Indeed, [14, Theorem 3.1 (ii)] shows that some Tits process J′ starting from E is still a subalgebra of J. Hence J′ is a Jordan division algebra of degree 3 and dimension 6 or 9 (see 2.1 below). But in dimension 6, such creatures do not exist since, otherwise, we would obtain a central associative division algebra of degree 3 with involution of the first kind, which is impossible ([4, (3.1)] and [1, V Theorem 17]). Hence J′ = J, and the extension theorem follows.

2. Proof of the extension theorem

Before starting with the proof, we require a few additional technicalities most of which are well known.

2.1 Springer forms. Let J = J(V, N, ♯, 1) be the Jordan algebra of a cubic norm structure over k as in 1.1. Given a cubic étale subalgebra E ⊂ J, we obtain an orthogonal decomposition J = E ⊕ E^{⊥} relative to T, and the assignment

(16)  (v, x) ⟼ v.x := −v × x  (v ∈ E, x ∈ E^{⊥})

endows E^{⊥} with the structure of a left E-module [11, Proposition 2.1 a)]. Furthermore, for x ∈ E^{⊥} we are allowed to write

(17)  x^{♯} = −q_{E}(x) + r_{E}(x)  (q_{E}(x) ∈ E, r_{E}(x) ∈ E^{⊥}),

and the map q_{E} : E^{⊥} ⟶ E is a quadratic form over E [11, Proposition 2.1 b)], called the Springer form of E in J. Notice that our definition of q_{E} differs from the one in [11], [14] by a sign, bringing us back to the original normalization due to Springer, cf. Springer-Veldkamp [17, (6.5)]. Thanks to (2), (16) we have

T (v.x, y) = T (x, v.y) = −T (v, x × y) (18) for all v ∈ E, x, y ∈ E^{⊥}. Furthermore, (4) implies

N(v.x) = N_{E}(v)N(x). (19)

Finally, by [14, Lemma 3.3, (ii): d), e), h) and (vi)],

(20)  r_{E}(v.x) = v^{♯}.r_{E}(x),
(21)  q_{E}(x, r_{E}(x)) = N(x)1,
(22)  r_{E}(r_{E}(x)) = N(x)x − q_{E}(x).r_{E}(x),
(23)  r_{E}(x, v.r_{E}(x)) = (v × q_{E}(x)).x.

2.2 Lemma. Notations being as in 2.1, we obtain

r_{E}(v.x + w.r_{E}(x)) = (N(x)w^{♯} + [w × (vq_{E}(x))]) .x + (v^{♯} − w^{♯}q_{E}(x)).r_{E}(x)

for all v, w ∈ E, x ∈ E^{⊥}.

Proof. For v = 1, the assertion follows by expanding the left-hand side and applying (20), (22), (23). In general, we may extend scalars if necessary and invoke Zariski density to assume v ∈ E^{×}. But then the special case v = 1 yields

r_{E}(v.x + w.r_{E}(x)) = r_{E}(v.[x + (v^{−1}w).r_{E}(x)])
= v^{♯}.r_{E}(x + (v^{−1}w).r_{E}(x))  (by (20))
= v^{♯}.[(N(x)(v^{−1}w)^{♯} + (v^{−1}w) × q_{E}(x)).x
+ (1 − (v^{−1}w)^{♯}q_{E}(x)).r_{E}(x)].

Here we apply the identity (st)^{♯} = s^{♯}t^{♯} and its linearization (st_{1}) × (st_{2}) = s^{♯}(t_{1} × t_{2}), both valid in E, to arrive at the desired conclusion. □

2.3 Associates of quadratic forms with base point. Let q : V ⟶ k be a quadratic form over k which is nonsingular in the sense that its induced symmetric bilinear form is nondegenerate, and suppose e ∈ V is a base point for q, so q(e) = 1. Denote by k^{•} the multiplicative group k^{×} for char k ≠ 2 and the additive group k for char k = 2. Given δ ∈ H^{1}(k, ℤ/2ℤ), choose any representative d ∈ k^{•} of δ to define q_{δ} : V ⟶ k by q_{δ}(x) = dq(x) + \frac{1−d}{4}q(e, x)^{2} (char k ≠ 2), q_{δ}(x) = q(x) + dq(e, x)^{2}(char k = 2). Then q_{δ} is a nonsingular quadratic form with base point e whose isometry class neither depends on d nor on e. This follows from Witt’s Theorem and the proof of [11, Proposition 3.1], although for char k ≠ 2, q_{δ} as defined here differs from the corresponding notion in [11] by a factor d. We call q_{δ} the δ-associate of q. This concept applies in particular to the norm of a quadratic étale algebra. From [15, 2.9] we recall

(N_{k{δ}})_{δ′} ≅ N_{k{δ+δ′}} (24)

for δ, δ′ ∈ H^{1}(k, ℤ/2ℤ).

2.4 We are now ready to prove the extension theorem 1.6, making the adjustments described in 1.7, and begin by reducing to the case that L is split. So let us assume for the time being that this case has been settled and suppose L/k is a separable quadratic field extension, with nontrivial k-automorphism σ. Extending scalars from k to L, we then conclude from 1.5 that, for some b ∈ L^{×}, the inclusion E ⊗ L ⊂ J ⊗ L extends to an isomorphism φ over L from the étale first Tits construction J(E ⊗ L, b) onto J ⊗ L. Using φ, the natural action of σ on J ⊗ L through the second factor will be transferred to J(E⊗L, b), making φ : J(E⊗L, b) ⟶ J ⊗L into a σ-equivariant map. Adapting the notations of 1.5 to the present set-up, we first observe that, since E ⊗ L ⊂ J(E ⊗ L, b) is stable under σ, so is its orthogonal complement

(E ⊗ L)^{⊥} = {(0, w_{1}, w_{2}) | w_{1}, w_{2} ∈ E ⊗ L} ⊂ J(E ⊗ L, b)

relative to T (cf. (13)). Hence

σ((0, 1, 0)) = (0, u_{1}, u_{2}), σ((0, 0, 1)) = (0, v_{1}, v_{2}) (25)

for some u_{1}, u_{2}, v_{1}, v_{2} ∈ E ⊗ L. Applying the norm to the first equation of (25) and using (11), we obtain

σ(b) = bN_{E}(u_{1}) + b^{−1}N_{E}(u_{2}). (26)

Similarly, applying the adjoint to the first equation of (25) and using (12), we obtain

u_{1}u_{2} = 0, v_{1} = N_{L}(b)^{−1}u_{2}^{♯}, v_{2} = bσ(b)^{−1}u_{1}^{♯}. (27)

Finally, applying σ to the relation σ((0, 1, 0)) = −u_{1} × (0, 1, 0) − u_{2} × (0, 0, 1) (by (25), (14)) and using (25), (27), we obtain

u_{1}σ(u_{1}) + N_{L}(b)^{−1}σ(u_{2})u_{2}^{♯} = 1, σ(u_{1})u_{2} + bσ(b)^{−1}u_{1}^{♯}σ(u_{2}) = 0. (28)

Next we show that u_{2} ∈ E ⊗L is invertible. Arguing indirectly, let us assume N_{E}(u_{2}) = 0. Then (26) reduces to σ(b) = bN_{E}(u_{1}), forcing u_{1} ∈ (E ⊗ L)^{×}. Therefore (27) implies

u_{2} = v_{1} = 0, v_{2} = u_{1}^{−1}. (29)

Moreover, N_{L}(u_{1}) = 1 by (28), so Hilbert’s Theorem 90 yields an element y ∈ (E ⊗L)^{×} satisfying

u_{1} = yσ(y)^{−1}. (30)

Now (26), (29), (30) imply that α := N_{E}(y)b ∈ L^{×} remains fixed under σ and hence belongs to k. Furthermore, the map

ψ : J(E, α) ⊗ L ⟶ J(E ⊗ L, b)

sending (w_{0}, w_{1}, w_{2}) to (w_{0}, w_{1}y, y^{−1}w_{2}) is an isomorphism over L. Letting σ act on J(E, a) ⊗ L through the second factor, we deduce from

σ (ψ((w_{0}, w_{1}, w_{2}))) = σ((w_{0}, w_{1}y, y^{−1}w_{2}))

= σ(w_{0} − w_{1}y × (0, 1, 0) − y^{−1}w_{2} × (0, 0, 1)) (by (14))

= σ(w_{0}) − σ(w_{1})σ(y) × (0, u_{1}, 0) − σ(y)^{−1}σ(w_{2}) × (0, 0, u_{1}^{−1}) (by (25), (29))

= (σ(w_{0}), σ(w_{1})y, y^{−1}σ(w_{2})) (by (14), (30))

= ψ (σ((w_{0}, w_{1}, w_{2})))

for all w_{0}, w_{1}, w_{2} ∈ E ⊗ L that ψ is σ-equivariant. Hence so is φ ∘ ψ : J(E, α) ⊗ L ⟶ J ⊗L, and restricting to the fixed points of σ gives an isomorphism J(E, a) ˜ ⟶ J. From 1.4, 1.5 and (15) we therefore conclude δ(L/k) = 0, so L is split. This contradiction shows that u_{2} ∈ E ⊗ L is indeed invertible. Hence (27), (26) collapse to

u_{1} = v_{2} = 0, N_{E}(u_{2}) = N_{L}(b), v_{1} = u_{2}^{−1}. (31)

Now (28) implies 1 = N_{L}(b)^{−1}σ(u_{2})u_{2}^{♯} = σ(u_{2})u_{2}^{−1}, so u := u_{2} belongs to E and satisfies N_{E}(u) = N_{L}(b). Furthermore, by (25), (31), σ((0, 1, 0)) = (0, 0, u), σ((0, 0, 1)) = (0, u^{−1}, 0). From this and (14) one easily concludes

σ((w_{0}, w_{1}, w_{2})) = (σ(w_{0}), σ(w_{2})u^{−1}, uσ(w_{1}))

for w_{0}, w_{1}, w_{2} ∈ E ⊗ L. Identifying E ⊂ E ⊗ L, J ⊂ J ⊗ L canonically and arguing as in the proof of [7, Theorem 9], it now follows that the assignment (w, x) ⟼ φ((w, x, uσ(x)) for w ∈ E, x ∈ E ⊗ L gives an isomorphism from the étale Tits process J(E, L, u, b) onto J extending the inclusion E ⊂ J.

2.5 In order to complete the proof, the reduction carried out in 2.4 allows us to assume that L ≅ k × k is split, which, by (15), implies

δ := δ(K/k) = δ(E/k) ∈ H^{1}(k, ℤ/2ℤ). (32)

In view of 1.5, we must show that, for some α ∈ k^{×}, the inclusion E ⊂ J extends to an isomorphism from the étale first Tits construction J(E, α) onto J. To do so, we employ the following specializations of results obtained in [11].

2.6 Proposition. ([11, Proposition 2.2]) Notations being as in 1.6, there exists a nonzero element α ∈ k such that the inclusion E ⊂ J extends to an isomorphism from the étale first Tits construction J(E, α) onto J if and only if some x ∈ E^{⊥}∩J^{×} satisfies q_{E}(x) = 0. □

2.7 Theorem. ([11, Theorem 3.2]) Notations being as in 1.6 and setting δ := δ(E/k), the Springer form of E in J is similar to (N_{K⊗E/E})_{δ} or (N_{K})_{δ} ⊗ E according as E is a field or not. □

2.8 Returning to 2.5, we now argue indirectly and assume that no α ∈ k^{×} allows an isomorphism J(E, α) ˜ ⟶ J extending the inclusion E ⊂ J. Then 2.6 implies

q_{E}(x) ≠ 0 for all x ∈ E^{⊥} ∩ J^{×}. (33)

Moreover, it follows from 2.7, (24) and (32) that the binary quadratic space (E^{⊥}, q_{E}) over E is the hyperbolic plane. Accordingly, we choose a hyperbolic pair (x_{1}, x_{2}) of (E^{⊥}, q_{E}), so E^{⊥} is a free E-module of rank 2 with basis (x_{1}, x_{2}) and

q_{E}(x_{1}) = q_{E}(x_{2}) = 0, q_{E}(x_{1}, x_{2}) = 1. (34)

In particular, x_{1}, x_{2} cannot be invertible in J (by (33)), and (21) implies

(35)  q_{E}(x_{i}, r_{E}(x_{i})) = N(x_{i})1 = 0  (i = 1, 2).

Furthermore, (17), (34) imply x_{i}^{♯} = r_{E}(x_{i}) = u_{i}.x_{i} + v_{i}.x_{3−i} for some u_{i}, v_{i} ∈ E, forcing v_{i} = 0 by (34), (35). Hence

(36)  x_{i}^{♯} = r_{E}(x_{i}) = u_{i}.x_{i}  (i = 1, 2).

We now claim

(37)  (v.x_{i})^{♯} = r_{E}(v.x_{i}) = u_{i}v^{♯}.x_{i}  (v ∈ E, i = 1, 2),
(38)  N_{E}(u_{i}) = 0  (i = 1, 2),
(39)  N(v_{1}.x_{1} + v_{2}.x_{2}) = T_{E}(u_{1}v_{1}^{♯}v_{2} + u_{2}v_{1}v_{2}^{♯})  (v_{i} ∈ E, i = 1, 2).

The first relation of (37) follows from q_{E}(v.x_{i}) = v^{2}q_{E}(x_{i}) = 0, the second one from (20), (36). Hence 0 = N(x_{i})x_{i} (by (35)) = x_{i}^{♯♯} (by (1)) = (u_{i}.x_{i})^{♯} (by (36)) = (u_{i}u_{i}^{♯}).x_{i} (by (37)) = N_{E}(u_{i})x_{i}, giving (38). This allows us to expand the left-hand side of (39):

N(v_{1}.x_{1} + v_{2}.x_{2}) = N(v_{1}.x_{1}) + T ((v_{1}.x_{1})^{♯}, v_{2}.x_{2}) + T (v_{1}.x_{1}, (v_{2}.x_{2})^{♯}) + N(v_{2}.x_{2})
= N_{E}(v_{1})N(x_{1}) + T (u_{1}v_{1}^{♯}.x_{1}, v_{2}.x_{2}) + T (v_{1}.x_{1}, u_{2}v_{2}^{♯}.x_{2})

+ N_{E}(v_{2})N(x_{2}) (by (19), (37))

= −T (u_{1}v_{1}^{♯}v_{2} + u_{2}v_{1}v_{2}^{♯}, x_{1} × x_{2}) (by (18), (35))

= T_{E}(u_{1}v_{1}^{♯}v_{2} + u_{2}v_{1}v_{2}^{♯}, q_{E}(x_{1}, x_{2})) (by (17)),

and (39) follows from (34).

Special Case. E ≅ k × k × k splits. Then δ = 0, and K ≅ k × k splits as well (by (32)). Hence J ≅ M_{3}(k)^{+}, and we may assume that E sits diagonally in J under this identification. But then

y := (\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}) ∈ E^{⊥} ∩ J^{×}

satisfies y^{♯} ∈ E^{⊥}, forcing q_{E}(y) = 0 and contradicting (33).

Returning to the general case, let us assume u_{1} = u_{2} = 0. Then (39) shows that N vanishes on E^{⊥} even after extending scalars to the algebraic closure. On the other hand, this extension brings us back to special case, where we have just produced invertible elements belonging to E^{⊥}. This is a contradiction, so by symmetry we may assume

u_{1} ≠ 0. (40)

Combining with (38), we conclude that E is not a field, forcing E ≅ k × K by (32). Hence we can find α_{i} ∈ k, a_{i} ∈ K satisfying u_{i} = (α_{i}, a_{i}) (i = 1, 2). Now recall from [13, Example 2.2] that the adjoint of E is given by

(41)  (β, b)^{♯} = (N_{K}(b), βτ(b))  (β ∈ k, b ∈ K)

and consider the idempotent c = (1, 0) ∈ E. From (34) we conclude q_{E}(c.x_{1} + (1 − c).x_{2}) = 0, and the relations c^{♯} = 0, (1 − c)^{♯} = c (by (41)) combine with (39) to imply N(c.x_{1} + (1 − c).x_{2}) = α_{2}. Hence α_{2} = 0 by (33). Interchanging x_{1}, x_{2}, we obtain α_{1} = 0 as well, so

(42)  u_{i} = (0, a_{i})  (i = 1, 2).

For v_{i} ∈ E, we next expand the expression r_{E}(v_{1}.x_{1} + v_{2}.r_{E}(x_{1})) in two different ways. On the one hand,

r_{E}(v_{1}.x_{1} + v_{2}.r_{E}(x_{1})) = v_{1}^{♯}.r_{E}(x_{1})  (by 2.2, (34), (35))
= u_{1}v_{1}^{♯}.x_{1}  (by (36)),

on the other

r_{E}(v_{1}.x_{1} + v_{2}.r_{E}(x_{1})) = r_{E}((v_{1} + u_{1}v_{2}).x_{1})  (by (36))
= u_{1}(v_{1} + u_{1}v_{2})^{♯}.x_{1}  (by (37))
= (u_{1}v_{1}^{♯} + u_{1}(v_{1} × u_{1}v_{2})).x_{1}

since u_{1}(u_{1}v_{2})^{♯} = u_{1}u_{1}^{♯}v_{2}^{♯} = 0 by (38). Hence u_{1}(v_{1} × u_{1}v_{2}) = 0. Using (41) to compute this explicitly for v_{1} = c = (1, 0), v_{2} = 1 − c = (0, 1), we obtain N_{K}(a_{1}) = 0. But a_{1} ≠ 0 by (40), (42), so K is not a field, and we are back again to the special case, a contradiction. □

3. An Isomorphism criterion

3.1 The general set-up. In this section we fix two étale Tits process algebras J = J(E, L, u, b), J′ = J(E′, L′, u′, b′) as in 1.3, where E, E′ and L, L′ are cubic and quadratic étale algebras, respectively, over our base field k and u ∈ E, u′ ∈ E′, b ∈ L, b′ ∈ L′ are invertible elements satisfying N_{E}(u) = N_{L}(b), N_{E′}(u′) = N_{L′}(b′). We denote by σ, σ′ the nontrivial k-automorphism of L, L′, respectively. All conventions of 1.3 remain in force. Letting φ : E′ ˜ ⟶E be any isomorphism, we will be concerned with the question as to when φ can be extended to an isomorphism from J′ onto J. A partial answer to this question may be found in the following result.

3.2 Theorem. Notations being as in 3.1, let η : J′ ⟶ J be an arbitrary map. Then the following statements are equivalent.

(i) η is an isomorphism extending φ.

(ii) There exist an isomorphism ψ : L′ ˜ ⟶L and an invertible element y ∈ E ⊗L such that

φ(u′) = N_{L}(y)u, ψ(b′) = N_{E}(y)b (43)

and

η((v′, x′)) = (φ(v′), y(φ ⊗ ψ)(x′)) (44)

for all v′ ∈ E′, x′ ∈ E′ ⊗ L′.

3.3 Proof of 3.2, (ii) ⟹ (i). This is the easy part. Since η obviously preserves units, it suffices to show that it preserves norms as well. But this follows directly from (6), (43), (44) and the fact that the isomorphism φ ⊗ ψ : E′ ⊗ L′ ⟶ E ⊗ L satisfies the relations

N_{E} ∘ (φ ⊗ ψ) = ψ ∘ N_{E′}, σ ∘ (φ ⊗ ψ) = (φ ⊗ ψ) ∘ σ′,
T_{E} ∘ ((φ ⊗ ψ) × (φ ⊗ ψ)) = ψ ∘ T_{E′}

on E′ ⊗ L′, (E′ ⊗ L′) × (E′ ⊗ L′), respectively. The converse implication (i) ⟹ (ii) requires a bit more work. First we have to deal with a side-issue.

3.4 Quadratic étale algebras over rings. Let R be a commutative associative ring of scalars. Although we are mostly concerned here with algebras over fields, the following digression to algebras over R cannot be avoided. So let A be a quadratic étale R-algebra with norm n, trace t and standard involution σ (cf. Knus [5, I (1.3), especially (1.3.6)]. Observe for the constant group scheme ℤ/2ℤ over ℤ (cf. Waterhouse [18, 2.3] that (ℤ/2ℤ)(R) identifies canonically with the group of idempotents in R under the composition (c, c′) ⟼ c + c′ − 2cc′ [5, III (4.1)]. Also, for c ∈ (ℤ/2ℤ)(R), the map

ψ_{c} : A ⟶ A, x ⟼ ψ_{c}(x) = cx + (1 − c)σ(x),

is an automorphism of A [5, III (4.1.2)].

3.5 Proposition. ([5, III (4.1.2)]) Notations being as in 3.4, the assignment c ⟼ ψ_{c} determines an isomorphism from (ℤ/2ℤ)(R) onto the automorphism group of A. □

3.6 Lemma. ([5, III (4.1.1)]) Notations being as in 3.4, suppose R is a local ring. Then A has a basis (1, θ) over R such that θ^{2} − θ + r1 = 0 for some r ∈ R satisfying 1 − 4r ∈ R^{×}. □

3.7 Lemma. Notations being as in 3.4, let ψ : A ⟶ A be an R-linear map preserving n and 1. Then ψ is an automorphism of A.

Proof. Since the question is local on R, we may assume that R is a local ring. Then A is generated by a single element (3.6), so it suffices to show that ψ preserves squares. But this follows immediately from n ∘ ψ = n, ψ(1) = 1, t ∘ ψ = t and the equation x^{2} − t(x)x + n(x)1 = 0 for all x ∈ A [5, I (1.3.3)]. □

3.8 Proof of 3.2, (i) ⟹ (ii). We proceed in several steps. 1^{0}. Let (B, τ), (B′, τ′) be central simple associative algebras of degree 3 with involution of the second kind satisfying J ≅ H(B, τ), J′ ≅ H(B′, τ′). Since (B′, τ′) is perfect (cf. Jacobson [3, Theorem 5]), η induces a unique isomorphism (B′, τ′) ˜ ⟶(B, τ) of algebras with involution. In particular, the centres of B, B′ are isomorphic, forcing δ(L/k) = δ(L′/k) by 1.4. Hence there exists an isomorphism ψ_{0} : L′ ˜ ⟶L. The étale Tits process being functorial in the obvious sense, we now obtain an isomorphism

η_{0} := φ × (φ ⊗ ψ_{0}) : J′ ˜ ⟶ J(E, L, φ(u′), ψ_{0}(b′)),

and after replacing η by η ∘ η_{0}^{−1}, we may assume E′ = E, φ = 1_{E}, L′ = L. 2^{0}. By 1^{0}, η induces the identity on E and hence matches the orthogonal complements of E relative to the trace forms of J, J′, repectively, which both identify canonically with E ⊗ L through the second factor (by (9)). We therefore obtain a k-linear bijection ψ : E ⊗ L ⟶ E ⊗ L satisfying

(45)  η((v, x)) = (v, ψ(x))  (v ∈ E, x ∈ E ⊗ L).

Using this an expanding η(v × (0, x)) = v × η((0, x)) by means of (10), we conclude that ψ is in fact E-linear. Similarly, expanding η((0, x)^{♯}) = η((0, x))^{♯} by means of (7) yields

N_{L}(ψ(x))u = N_{L}(x)u′, (46)

ψ (σ(b′)σ(x)^{♯}u′^{−1}) = σ(b)σ(ψ(x))^{♯}u^{−1} (47)

for all x ∈ E ⊗ L. 3^{0}. We put y := ψ(1 ⊗ 1) ∈ E ⊗ L and specialize x = 1 ⊗ 1 in (46) to conclude

u′ = N_{L}(y)u. (48)

In particular, y is invertible. Also, by 3.3, the assignment (v, x) ⟼ (v, y^{−1}x) gives an isomorphism

η_{1} : J ⟶ J(E, L, u_{1}, b_{1})

where u_{1} = u′, b_{1} = N_{E}(y)b. 4^{0}. Using 3^{0} and replacing η by η_{1} ∘ η if necessary, we may assume ψ(1 ⊗ 1) = 1 ⊗ 1. Then (48) reduces to

u′ = u, (49)

and (46) yields N_{L} ∘ ψ = N_{L}. Hence, by 3.7, ψ is an automorphism of the quadratic étale E-algebra E⊗L. (Notice that E need not be a field, so we do require the generality provided for by 3.4.) Now 3.5 yields an idempotent e ∈ E satisfying

(50)  ψ(x) = ex + (1 − e)σ(x)  (x ∈ E ⊗ L).

On the other hand, since ψ is E-linear, we obtain

u′^{−1}ψ(1 ⊗ σ(b′)) = ψ (u′^{−1} ⊗ σ(b′))
= ψ (σ(b′)σ(1 ⊗ 1)^{♯}u′^{−1})
= σ(b)σ(ψ(1 ⊗ 1))^{♯}u^{−1}  (by (47))
= u^{−1}(1 ⊗ σ(b)),

and (49) implies

σ(1 ⊗ b) = 1 ⊗ σ(b) = ψ(1 ⊗ σ(b′))
= e(1 ⊗ σ(b′)) + (1 − e)(1 ⊗ b′)  (by (50))
= σ(e ⊗ b′ + (1 − e) ⊗ σ(b′)).

Summing up, we have 1 ⊗ b = e ⊗ b′ + (1 − e) ⊗ σ(b′). (51) 5^{0}. By (50), (51), the proof will be complete once we have shown e = 0, 1. We argue indirectly and assume e ≠ 0, 1. Then (41) shows either e^{♯} = 0 or (1 − e)^{♯} = 0. Since the assignment (v, x) ⟼ (v, σ(x)) determines an automorphism η_{2} of J by 3.3, we may replace η by η_{2} ∘ η if necessary to obtain e^{♯} = 0, (1 − e)^{♯} = e. (52) Since e ⊗ 1, (1 − e) ⊗ 1 ∈ E ⊗ L are free over L by assumption, the relation

b′(e ⊗ 1) + σ(b′)((1 − e) ⊗ 1) = e ⊗ b′ + (1 − e) ⊗ σ(b′)
= 1 ⊗ b  (by (51))
= b(e ⊗ 1) + b((1 − e) ⊗ 1)

implies b = b′ ∈ k. (53) Given a ∈ L, we now to expand, using (52),

N_{E}(e ⊗ a + (1 − e) ⊗ σ(a)) = N_{E}(a(e ⊗ 1) + σ(a)((1 − e) ⊗ 1)))

= a^{3}N_{E}(e) + a^{2}σ(a)T_{E}(e^{♯}, 1 − e) + aσ(a)^{2}T_{E}(e, (1 − e)^{♯}) + σ(a)^{3}N_{E}(1 − e)
= aσ(a)^{2}

to obtain N_{E}(e ⊗ a + (1 − e) ⊗ σ(a)) = N_{L}(a)σ(a). (54) Next observe that η, being an isomorphism, preserves norms and hence satisfies N_{J} ∘η = N_{J′}. Applying both sides to (0, 1 ⊗ a), and using (45), (50), (6), (53), (54) we deduce

N_{J} ∘ η((0, 1 ⊗ a)) = N_{J}((0, e ⊗ a + (1 − e) ⊗ σ(a)))

= b [N_{E}(e ⊗ a + (1 − e) ⊗ σ(a)) + σ(N_{E}(e ⊗ a + (1 − e) ⊗ σ(a)))]

= T_{L}(a)N_{L}(a)b

on the one hand,

N_{J′}((0, 1 ⊗ a)) = b[N_{E}(1 ⊗ a) + σ(N_{E}(1 ⊗ a))]
= T_{L}(a^{3})b
= (T_{L}(a)^{3} − 3T_{L}(a)N_{L}(a))b

on the other. Comparing we conclude T_{L}(a)^{3} = 4T_{L}(a)N_{L}(a) for all a ∈ L. But 3.6 yields an element θ ∈ L satisfying T_{L}(θ) = 1, 1 − 4N_{L}(θ) ≠ 0, a contradiction. This completes the proof of 3.2. □

4. Isomorphic embeddings

4.1 Norm classes. In this section, we fix a central simple associative algebra (B, τ) of degree 3 with involution of the second kind over k and a cubic étale k-algebra E. As in 1.6, we write K for the centre of B, J = H(B, τ) for the Jordan algebra of τ-symmetric elements in B and L for the quadratic étale k-algebra corresponding to the element δ(E/k) + δ(K/k) ∈ H^{1}(k, ℤ/2ℤ). Given an isomorphic embedding ι : E ⟶ J, the extension theorem 1.6 yields invertible elements u ∈ E, b ∈ L satisfying N_{E}(u) = N_{L}(b) such that ι extends to an isomorphism form J(E, L, u, b) onto J. Applying 3.2 to φ = 1_{E}, it follows that

[ι] := u mod N_{L}((E ⊗ L)^{×}) ∈ E^{×}/N_{L}((E ⊗ L)^{×})

is independent of all choices made. We call [ι] the norm class of ι. Our main objective in this section is to prove the following result.

4.2 Theorem. Given isomorphic embeddings ι, ι′ : E ⟶ J, the following statements are equivalent.

(i) ι and ι′ have the same norm class, i.e., [ι] = [ι′].

(ii) ι and ι′ are equivalent, i.e., there exists an automorphism φ of J moving ι to ι′, so φ ∘ ι = ι′.

Since the norm class by 4.1 is well defined, (ii) obviously implies (i). To establish the converse, we argue similarly to the proof of the Skolem-Noether Theorem for ninedimensional subalgebras of Albert algebras due to Parimala-Sridharan-Thakur [9], see [4, (40.15)]. We begin by examining the étale first Tits construction 1.5.

4.3 Proposition. For α, α′ ∈ k^{×} the following statements are equivalent.

(i) J(E, α) and J(E, α′) are isomorphic.

(ii) J(E, α) and J(E, α′) are isotopic.

(iii) α ≡ α^{′ε} mod N_{E}(E^{×}) for some ε ∈ {±1}.

(iv) The identity of E can be extended to an isomorphism from J(E, α) to J(E, α′).

Proof. The implications (iv) ⟹ (i) ⟹ (ii) are clear. (iii) ⟹ (iv). We have α = α^{′ε}N_{E}(v) for some ε ∈ {±1} and v ∈ E^{×}. Hence, if ε = 1, the assignment (w_{0}, w_{1}, w_{2}) ⟼ (w_{0}, w_{1}v, v^{−1}w_{2}) gives an isomorphism of the desired kind. On the other hand, if ε = −1, the assignment (w_{0}, w_{1}, w_{2}) ⟼ (w_{0}, w_{2}v^{−1}, vw_{1}) gives an isomorphism of the desired kind (since E is commutative). (ii) ⟹ (iii). We distinguish the following cases.

Case 1. δ(E/k) = 0. Then E is either a cyclic cubic field extension or it splits. Furthermore, 1.4 yields central simple associative algebras A, A′ of degree 3 over k satisfing J(E, α) ≅ A^{+}, J(E, α′) ≅ A^{′+}, and (ii) allows us to assume A ≅ A′. Let Ψ : A′ ⟶^{∼} A be an isomorphism. Then we may either apply the classical Skolem-Theorem to E, Ψ(E) ⊂ A if E is a field or use standard facts about complete orthogonal systems of primitive idempotents in full matrix algebras otherwise to conclude (iv). Using the indentifications

J(E, α) = J(E, k × k, (1, 1), (α, α^{−1})), J(E, α′) = J(E, k × k, (1, 1), (α′, α^{′−1}))

guaranteed by 1.5, we may therefore apply 3.2 to φ = 1_{E} and obtain (iii).

Case 2. δ(E/k) ≠ 0. Changing scalars from k to M = k{δ(E/k)} brings us back to Case 1, so some ε ∈ {±1} has αα^{′ε} ∈ N_{E}((E ⊗ M)^{×}), forcing the norm of J(E, αα^{′ε}) to become isotropic over M (1.5). Hence it must have been isotropic to begin with [17, 4.2.11], and (iii) follows. □

4.4 Remark. For char k ≠ 2, 3, the preceding result can also be derived by using the cohomological invariants attached to associative algebras of degree 3 [4, §30.C]. Indeed, given E, α as in 4.3, E for simplicity assumed to be a field, write J(E, α) = H(B_{α}, τ_{α}) for some central simple associative algebra (B_{α}, τ_{α}) of degree 3 with involution of the second kind. Then the centre of B_{α} agrees with the discriminant algebra of E (1.4), so f_{1}(B_{α}, τ_{α}) ∈ H^{1}(k, ℤ/2ℤ) is independent of α. Furthermore, τ_{α} is distinguished [4, (39.5)(3)], forcing f_{3}(B_{α}, τ_{α}) = 0. Finally, if K is a field and ρ denotes a fixed nontrivial K-automorphism of E ⊗ K, we obtain B_{α} ≅ (E, ρ, α) or B_{α} ≅ (E, ρ, α^{−1}) as cyclic Kalgebras of degree 3, and g_{2}(B_{α}, τ_{α}) is uniquely determined by the Brauer class of B_{α}; on the other hand, if K ≅ k × k splits, E is cyclic, so fixing a nontrivial k-automorphism ρ of E, we obtain B_{α} ≅ A_{α} × A_{α}^{op} where A_{α} ≅ (E, ρ, α) or A_{α} ≅ (E, ρ, α^{−1}), and g_{2}(B_{α}, τ_{α}) is uniquely determined by the Brauer class of A_{α}. Summing up, 4.3 now follows from the fact that central simple associative algebras of degree 3 with involutions of the second kind are classified by their invariants f_{1}, f_{3}, g_{2} [4, (30.21)]. □

The following technicality is a variant of [4, (40.13)] over fields of arbitrary characteristic. We keep the notational conventions of 1.3.

4.5 Lemma. Let L, E be étale k-algebras of dimension 2, 3 respectively. Given y ∈ E ⊗ L such that c := N_{E}(y) ∈ L satisfies N_{L}(c) = 1, there exists an element y′ ∈ E ⊗ L satisfying N_{E}(y′) = c, N_{L}(y′) = 1.

Proof. By Hilbert’s Theorem 90, we have c = dσ(d)^{−1} for some d ∈ L^{×}. If E is not a field, N_{E} is surjective, so some z ∈ E ⊗ L has N_{E}(z) = d, and y′ = zσ(z)^{−1} does the job. Hence we may assume that E is a field. Choose θ ∈ L − k, κ ∈ L^{×} satisfying θ + σ(θ) = 1, σ(κ) = −κ. Then the map F : E × k ⟶ k defined by

F ((x, ξ)) := κ(σ(d)N_{E}(x + ξθ1) − dN_{E}(x + ξ(1 − θ)1))

for x ∈ E, ξ ∈ k is a cubic form. We now distinguish the following cases.

Case 1. L ≅ k × k splits. Then E ⊗ L = E × E, y = (y_{1}, y_{2}), y_{i} ∈ E^{×}(i = 1, 2) and c = (γ, γ^{−1}), γ = N_{E}(y_{1}), so the lemma follows by setting y′ = (y_{1}, y_{1}^{−1}). We also claim that F is isotropic. To see this, we may assume d = (γ, 1) and have θ = (α, 1 − α) for some α ∈ k, 2α ≠ 1. Since y_{1} = 1 implies d = (1, 1), hence F ((1, 0)) = 0, we may assume y_{1} ≠ 0. Then a direct computation shows that x := (y_{1} − 1)^{−1}(α1 + (α − 1)y_{1}) ∈ E satisfies F ((x, 1)) = 0, proving our claim.

Case 2. L is a field. Since by Case 1, the cubic form F becomes isotropic after changing scalars from k to L, it must have been so all along [17, 4.2.11]. Hence there exists a nonzero element (x, ξ) ∈ E × k satisfying

σ(d)N_{E}(x + ξθ1) = dN_{E}(x + ξ(1 − θ)1).

This implies x + ξθ1 ∈ (E ⊗ L)^{×}, and the lemma follows by setting

y′ := (x + ξθ1)σ(x + ξθ1)^{−1}.  □

4.6 Remark. A similar argument will give a proof of [4, (40.13)] in all characteristics, see [10].

4.7 Proof of 4.2, (i) ⟹ (ii). Keeping the notations of 4.1, we apply 1.6 to obtain invertible elements u, u′ ∈ E, b, b′ ∈ L satisfying N_{E}(u) = N_{L}(b), N_{E}(u′) = N_{L}(b′) such that ι, ι′ extend to isomorphisms

η : J_{1} := J(E, L, u, b) ⟶^{∼} J, η′ : J′_{1} = J(E, L, u′, b′) ⟶^{∼} J,

respectively; in particular, J_{1} ≅ J′_{1} under η^{′−1} ∘ η. We now distinguish the following cases.

Case 1. L ≅ k × k splits. Then we may identfy J_{1} = J(E, α), J′_{1} = J(E, α′) for some α, α′ ∈ k^{×}, and 4.3 yields an isomorphism ϕ : J_{1} ⟶^{∼} J′_{1} which is the identity on E. Hence φ := η′ ∘ ϕ ∘ η^{−1} ∈ Aut(J) satisfies φ ∘ ι = ι′, forcing ι, ι′ to be equivalent.

Case 2. L is a field. Changing scalars from k to L, J_{1}⊗L ≅ J(E⊗L, b) and J′_{1}⊗L ≅ J(E⊗L, b′) [13, 3.5, 3.8] continue to be isomorphic, so 4.3 yields b = b′^{ε}N_{E}(z) for some ε ∈ {±1}, z ∈ (E ⊗L)^{×}. Applying 3.2 for E = E′, L = L′, φ = 1_{E}, ψ = σ, u = u′^{−1}, b = b′^{−1}, y = u′ we may in fact assume ε = 1, so b = b′N_{E}(z). Applying 3.2 again, this time for E = E′, L = L′, φ = 1_{E}, ψ = 1_{L}, y = z^{−1} we even reduce to the case z = 1, so b = b′. Now we use the fact that ι and ι′ have the same norm class, so u′ = N_{L}(y)u for some invertible element y ∈ E ⊗ L. This implies N_{L}(c) = 1 for c = N_{E}(y) ∈ L, and by 4.5 we obtain an element y′ ∈ E ⊗ L such that N_{E}(y′) = N_{E}(y), N_{L}(y′) = 1. Hence the assignment (v, x) ⟼ (v, y′^{−1}yx) gives an isomorphism ϕ : J_{1} ⟶^{∼} J′_{1} which is the identity on E (3.2), so φ := η′ ∘ ϕ ∘ η^{−1} ∈ Aut(J) moves ι to ι′, and the proof is complete. □

4.8 Remark. If K is isomorphic to the discriminant algebra of E, then L ≅ k×k must be split, forcing the norm class of group E^{×}/N_{L}((E ⊗L)^{×}) to be trivial. Hence, thanks to 4.2, any two embeddings from E to J are equivalent. According to the discussion of Case 1 in the preceding proof, this statement amounts to exactly the same as 4.3. □

4.9 Example. We now specialize 4.1 to the case that E = k × k × k is split. Giving an isomorphic embedding ι : E ⟶ J amounts to the same as giving a complete orthogonal system (e_{1}, e_{2}, e_{3}) of absolutely primitive idempotents in J. This in turn leads to a matrix g = diag(γ_{1}, γ_{2}, γ_{3}) ∈ GL_{3}(k) and an identification J = H_{3}(K, g), the Jordan algebra g-hermitian 3-by-3 matrices x having entries in K (so x = g^{−1}x^{∗}g, x^{∗} the conjugate thranspose of x), which matches e_{1}, e_{2}, e_{3} with the diagonal idempotents in H_{3}(K, g). We may clearly assume det g = 1, so u = (γ_{1}, γ_{2}, γ_{3}) ∈ E has norm 1. It now follows easily that the linear bijection

ϕ : J(E, K, u, 1) ⟶ J = H_{3}(K, g)

defined by

ϕ((v, y)) := (\begin{matrix}α_{1} & γ_{3}γ_{2}a_{3} & γ_{2}γ_{3}σ(a_{2}) \\ γ_{3}γ_{1}σ(a_{3}) & α_{2} & γ_{1}γ_{3}a_{1} \\ γ_{2}γ_{1}a_{2} & γ_{1}γ_{2}σ(a_{1}) & α_{3} \end{matrix})

for v = (α_{1}, α_{2}, α_{3}) ∈ E, y = (a_{1}, a_{2}, a_{3}) ∈ E ⊗ K = K × K × K, preserves norms and units, hence is an isomorphism extending ι. Making the obvious identifications and writing [α] for the coset of α ∈ k^{×} in k^{×}/N_{K}(K^{×}), we conclude

[ι] = ([γ_{1}], [γ_{2}], [γ_{3}]) = ([γ_{2}^{−1}γ_{3}], [γ_{3}^{−1}γ_{1}], [γ_{1}^{−1}γ_{2}]).

Hence the norm classes of isomorphic embeddings from E to ninedimensional absolutely simple Jordan algebras of degree 3 are exactly analogous to the norm classes of complete orthogonal systems of absolutely primitive idempotents in Albert algebras [2].

4.10 Example. We close the paper by constructing isomorphic embeddings that have distinct norm classes and hence are not equivalent (4.2). Let k_{0} be a field, for simplicity assumed to be of characteristic not two, and K_{0}/k_{0} a quadratic field extension. Given an indeterminate X, we put k = k_{0}(X), K = K_{0}(X),

g = diag(1, X, X^{−1}), g′ = diag((X + 1)^{−1}, X, X^{−1}(X + 1)),

and J = H_{3}(K, g), J′ = H_{3}(K, g′). Since X + 1 is represented by the 2-fold Pfister form N_{K}⊥ < X > N_{K}, we conclude that

N_{K}⊥ < X^{−1} > N_{K}⊥ < X > N_{K}

and

< X + 1 > N_{K}⊥ < X^{−1} > N_{K}⊥ < (X + 1)^{−1}X > N_{K}

are isometric, forcing J and J′ to have isometric trace forms [6, p. 502]. Hence they are isomorphic [4, (19.6)]. Now let ϕ : J′ ⟶^{∼} J be any isomorphism, put E = k × k × k and denote by ι (resp. ι′) the isomorphic embedding E ⟶ J corresponding to the diagonal idempotents in J (resp. the image under ϕ of the diagonal idempotents in J′). Then 4.9 implies

[ι] = ([1], [X], [X]), [ι′] = ([X + 1], [X], [X(X + 1)]).

In particular, [ι] ≠ [ι′] since, otherwise, we would obtain X + 1 ∈ N_{K}(K^{×}), which is easily seen to be impossible (for example, K is unramified at X). A similar argument shows X ≢ (X+1)mod N_{K}(K^{×}), so no embedding E ⟶ J arising from ι by composing it with an automorphism of E can be equivalent to ι′. It would be interesting to have an example of this kind where E or even J as in 4.1 are division algebras. □

References

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