On derivation algebras of some exceptional Jordan algebras in

characteristic two.

Pablo Alberca Bjerregaard^{∗} Departamento de Matemática Aplicada University of Málaga. 29012. Málaga. Spain

Cándido Martín González^{†}

Departamento de Álgebra, Geometría y Topología. University of Málaga, Apdo. 59, 29080, Málaga. Spain.

August 19, 2001

Abstract

The Chevalley-Schafer theorem ([7]) affirms that the derivation Lie algebra of the exceptional Jordan algebra H_{3}(C, −) is simple if the characteristic is neither two nor three. If the characteristic of the base field is 3 the result is also true (see [3]). In this one we describe, by using quadratic Jordan algebras, the situation in the split characteristic two case: the derivation Lie algebra is not simple. Moreover, there exists only one proper nonzero ideal I which is 26-dimensional, simple as a Lie algebra and such that the quotient algebra Der(H_{3}(C, −))/I is also simple and isomorphic to I.

1 Definitions and results

A quadratic Jordan algebra with 1 over an arbitrary commutative ring Φ with 1 is a triple (J, U, 1) where J is a unital left Φ-module, 1 a distinguished element of J, and U is a mapping a ↦ U_{a} of J into Hom_{Φ}(J, J) satisfying the following axioms: i. U is Φ-quadratic, that is, U_{αa} = α^{2}U_{a}, α ∈ Φ, a ∈ J and U_{a,b} := U_{a+b} −

∗E-Mail: pgalberca@uma.es †E-Mail: candido@apncs.cie.uma.es Both authors are supported by the project FQM-0125 of the Junta de Andalucía III Plan Andaluz de Investigación, by the spanish AECI project ’Estructuras algebraicas asociativas y no asociativas’, by the project ’Estudio analítico-algebraico de Sistemas Triples y de Pares en diferentes clases de estructuras no asociativas’ of the Junta de Andalucía and by the DGICYT project PB97-1497.

U_{a} − U_{b} is Φ-bilinear in a and b.

ii. U_{1} = 1.

iii. U_{a}U_{b}U_{a} = U_{U_{a}(b)}.

iv. U_{x}(T (y, x, z)) = T (x, y, U_{x}(z)), ∀x, y, z ∈ J where T (y, x, z) := U_{y,z}(x).

v. Conditions 3 and 4 hold in every scalar extension of J.

If (O_{s}, −) is a split Cayley algebra (an split octonions algebra) over F with − as its involution, then H_{3}(O_{s}, −) is the set of matrices

X = ( \begin{matrix}λ_{1} & a & b \\ \overline{a} & λ_{2} & c \\ \overline{b} & \overline{c} & λ_{3} \end{matrix} ) , with λ_{i} ∈ F, a, b, c ∈ O_{s}.  (1)

It is usual to express the generic element (1) of H_{3}(O_{s}, −) as

X = λ_{1}E_{1} + λ_{2}E_{2} + λ_{3}E_{3} + X_{1}(a) + X_{2}(b) + X_{3}(c).  (2)

where the definitions of E_{i} and X_{i} are the obvious ones. A basis of H_{3}(O_{s}, −) is the following

B := {E_{1}, E_{2}, E_{3}, X_{1}(e_{i})_{i=1,...,8}, X_{2}(e_{i})_{i=1,...,8}, X_{3}(e_{i})_{i=1,...,8}},  (3)

where e_{i}, with i = 1, …, 8, is the standard basis of the alternative algebra O_{s}, seen as the Zorn matrices algebra, that is,

e_{1} := ( \begin{matrix}1 & (0, 0, 0) \\ (0, 0, 0) & 0 \end{matrix} ) , e_{2} := ( \begin{matrix}0 & (0, 0, 0) \\ (0, 0, 0) & 1 \end{matrix} ) ,

e_{3} := ( \begin{matrix}0 & (1, 0, 0) \\ (0, 0, 0) & 0 \end{matrix} ) , e_{4} := ( \begin{matrix}0 & (0, 1, 0) \\ (0, 0, 0) & 0 \end{matrix} ) ,

e_{5} := ( \begin{matrix}0 & (0, 0, 1) \\ (0, 0, 0) & 0 \end{matrix} ) , e_{6} := ( \begin{matrix}0 & (0, 0, 0) \\ (1, 0, 0) & 0 \end{matrix} ) ,

e_{7} := ( \begin{matrix}0 & (0, 0, 0) \\ (0, 1, 0) & 0 \end{matrix} ) , e_{8} := ( \begin{matrix}0 & (0, 0, 0) \\ (0, 0, 1) & 0 \end{matrix} ) .

In order to construct a quadratic Jordan structure in H_{3}(O_{s}, −) we use McCrimmon’s equations ([3]):

U_{a[ii]}b[ii] = aba[ii]  (4)
U_{a[ij]}b[ii] = \overline{a}ba[jj]  (5)
U_{a[ij]}b[ij] = a\overline{b}a[ij]  (6)
T (a[ii], b[ij], c[jj]) = abc[ij]  (7)

T (a[ii], b[ij], c[ji]) = (abc + \overline{abc})[ii]  (8)
T (a[ii], b[ij], c[jk]) = abc[ik]  (9)
T (a[ii], b[ii], c[ij]) = abc[ij]  (10)
T (a[ij], b[jj], c[jk]) = abc[ik]  (11)
T (a[ij], b[ji], c[ik]) = a(bc)[ik]  (12)
T (a[ij], b[jk], c[ki]) = (a(bc) + \overline{a(bc)})[ii]  (13)

where a[ij] represents an element of H_{3}(C, −) filled with zeros except for an octonion a at the (i, j) position, T (a, b, c) = U_{a,c}(b) and it is understood that all the U formulas not covered by these and a[ji] = \overline{a}[ij] are 0. A derivation in this context (see [4, 1.4,p.3]) is a linear map D satisfying:

D(1) = 0,  (14)
D(U_{x}(y)) = T (D(x), y, x) + U_{x}(D(y)), ∀x, y.  (15)

It is easy to see that the vector space of all derivations of H_{3}(O_{s}, −) is a Lie algebra with respect to the Lie product [D, D′] := DD′ − D′D. Then we write f_{4}(O_{s}, −) to be the Lie algebra of the derivations of the quadratic Jordan algebra H_{3}(O_{s}, −). It is important to note that this construction does not depend on the characteristic of the base field.

2 Generic element of f_{4}(O_{s}, −) without any restriction of the characteristic

We have implemented the structure of H_{3}(O_{s}, −) into a computer system and, by using the multiplications rules with U and T and the conditions that define a derivation, we have determined the generic matrix of a derivation in f_{4}(O_{s}, −) in the basis B (see [2]). Let us see a couple of examples. We have put the image of E_{1} by an arbitrary derivation D to be a generic element of H_{3}(O_{s}, −) with 27 free parameters. As E_{1} is idempotent, that is, U_{E_{1}}(1) = E_{1}, we have that D(E_{1}) = D(U_{E_{1}}(1)) = T (D(E_{1}), 1, E_{1}) + U_{E_{1}}(D(1)) = T (D(E_{1}), 1, E_{1}). This implies that D(E_{1}) − T (D(E_{1}), 1, E_{1}) = 0 and solving the linear system given by this, we find that D(E_{1}) can be written in terms of 16 free param-

eters (two octonions) as D(E_{1}) = ( \begin{matrix}0 & O_{1} & O_{2} \\ \overline{O_{1}} & 0 & 0 \\ \overline{O_{2}} & 0 & 0 \end{matrix} ). In order to show the

analogous calculus with X_{1}(e_{1}), we have defined D(X_{1}(e_{1})) to be a generic element of H_{3}(O_{s}, −) and then we have used relations as U_{X_{1}(e_{1})}(E_{1}) = 0, to obtain the corresponding one after taking D, that is, T (D(X_{1}(e_{1})), E_{1}, X_{1}(e_{1}))+ U_{X_{1}(e_{1})}(D(E_{1})) = 0. Since T (E_{1}, E_{1}, X_{1}(e_{1})) = X_{1}(e_{1}), we have D(X_{1}(e_{1})) = T (D(E_{1}), E_{1}, X_{1}(e_{1}))+T (E_{1}, D(E_{1}), X_{1}(e_{1}))+T (E_{1}, E_{1}, D(X_{1}(e_{1}))), where we have used the Leibnitz rule.

Following this kind of calculus, we have obtained (see [2]) that the generic element of f_{4}(O_{s}, −), without any restriction on the characteristic, is as (55) with dim f_{4}(O_{s}, −) = 52.

3 Characteristic two case

We can now, by using the matrix (55), define a basis of f_{4}(O_{s}, −) in the characteristic two case,

B := {x_{i} : i = 1, …, 52}.  (16)

We have made this by sorting the parameters in (55)

{ξ, α_{1}, α_{2}, α_{3}, α_{4}, α_{5}, α_{6}, α_{7}, α_{8}, β_{1}, β_{2}, β_{3}, β_{4}, β_{5}, β_{6}, β_{7}, β_{8},
γ_{1}, γ_{2}, γ_{3}, γ_{4}, γ_{5}, γ_{6}, γ_{7}, γ_{8}, δ_{1}, δ_{2}, δ_{3}, δ_{4}, δ_{5}, δ_{6}, δ_{7}, ϵ_{1}, ϵ_{2}, ϵ_{3}, ϵ_{4},
η_{1}, η_{2}, η_{3}, η_{4}, η_{5}, ρ_{1}, ρ_{2}, ρ_{3}, ρ_{4}, ρ_{5}, ρ_{6}, ϕ_{1}, ϕ_{2}, χ_{1}, ψ_{1}, ψ_{2}}  (17)

and making them to be one and zero the rest in each case.

Let us now study the ideal generated by, for example, x_{1}. If we make the products [x_{1}, x_{i}], i = 1, …, 52, with all the elements of the basis B we obtain^{1} the linear independent set

{x_{1}, x_{10}, x_{11}, x_{12}, x_{13}, x_{14}, x_{15}, x_{16}, x_{17}, x_{18}, x_{19}, x_{20}, x_{21}, x_{22}, x_{23}, x_{24}, x_{25}}

(18)

where we have eliminated the zero elements and included x_{1}. By making products again of the elements in the previous set with all the basic elements we obtain the bigger linear independent set

{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}, x_{11}, x_{12}, x_{13}, x_{14}, x_{15}, x_{16}, x_{17}, x_{18},

x_{19}, x_{20}, x_{21}, x_{22}, x_{23}, x_{24}, x_{25}, x_{26} + x_{34} + x_{37}}. (19)

So the elements of the previous set are all the elements [[x_{1}, x_{i}], x_{j}] or [x, x_{i}] eliminating zeros and including x_{1}. If we increase the depth of the Lie products, we obtain the same set, that is, the ideal I generated by x_{1} is not the whole Lie algebra: the Lie algebra f_{4}(O_{s}, −) is not simple if the characteristic of the base field is two. If we proceed this way, we obtain that the ideal generated by any of the first 25 elements of the basis B is also I, and the remaining ones generate the whole algebra.

It will be very useful the following theorem:

Theorem 1 Let f_{4}(O_{s}, −) be the Lie algebra of the derivation of the quadratic Jordan algebra H_{3}(O_{s}, −) without any restriction on the characteristic. Then for all 0 ≠ X ∈ f_{4}(O_{s}, −) there exists i ∈ {1, …, 52} such that x_{i} ∈ (X), where (X) denotes the ideal generated by X.

Proof.

If X is a generic element of f_{4}(O_{s}, −), as in (55), we have the identity [[[X, x_{2}], x_{11}], x_{45}] = δ_{2}x_{19}. If δ_{2} ≠ 0, the basic element x_{19}, and the ideal generated by x_{19}, that is, I, is contained in the ideal generated by X. If δ_{2} = 0, we use the identity [[X, x_{38}], x_{45}] = −ϵ_{1}x_{45} to have that x_{45} ∈ (X) if ϵ_{1} ≠ 0.

1The characteristic of the base field is two.

If δ_{2} and ϵ_{1} are zero then [[X, x_{28}], x_{38}] = −ρ_{5}x_{38}. This equation allows us to conclude that x_{38} ∈ (X) if ρ_{5} ≠ 0. All the rest identities are

If also ρ_{5} =0 then [[X, x_{27}], x_{28}]=−ρ_{4}x_{28}; if also ρ_{4} =0 then [[X, x_{27}], x_{29}]=−ρ_{6}x_{27};
if also ρ_{6} =0 then [[X, x_{27}], x_{31}]=−ρ_{2}x_{27}; if also ρ_{2} =0 then [[X, x_{27}], x_{32}]=−ρ_{3}x_{27};
if also ρ_{3} =0 then [[Z, x_{27}], x_{30}]=α_{1}x_{3}; if also α_{1} =0 then [[X, x_{3}], x_{30}]=−ρ_{1}x_{3};
if also ρ_{1} =0 then [[X, x_{27}], x_{33}]=−η_{2}x_{27}; if also η_{2} =0 then [[X, x_{27}], x_{48}]=−η_{3}x_{27};
if also η_{3} =0 then [[X, x_{27}], x_{36}]=−γ_{6}x_{20}; if also γ_{6} =0 then [[X, x_{20}], x_{36}]=χ_{1}x_{20};
if also χ_{1} =0 then [[X, x_{27}], x_{51}]=η_{4}x_{27}; if also η_{4} =0 then [[X, x_{27}], x_{49}]=β_{5}x_{17};
if also β_{5} =0 then [[X, x_{17}], x_{49}]=−ϵ_{3}x_{17}; if also ϵ_{3} =0 then [[X, x_{26}], x_{27}]=η_{5}x_{32};
if also η_{5} =0 then [[X, x_{27}], x_{34}]=β_{4}x_{17}; if also β_{4} =0 then [[X, x_{1}], x_{27}]=γ_{2}x_{20};
if also γ_{2} =0 then [[X, x_{10}], x_{27}]=−α_{6}x_{18}; if also α_{6} =0 then [[X, x_{4}], x_{42}]=−δ_{5}x_{4};
if also δ_{5} =0 then [[X, x_{16}], x_{35}]=−ϕ_{2}x_{16}; if also ϕ_{2} =0 then [[X, x_{18}], x_{50}]=ϵ_{4}x_{18};
if also ϵ_{4} =0 then [[X, x_{26}], x_{28}]=−γ_{7}x_{18}; if also γ_{7} =0 then [[X, x_{1}], x_{28}]=β_{3}x_{17};
if also β_{3} =0 then [[X, x_{10}], x_{28}]=−α_{7}x_{18}; if also α_{7} =0 then [[X, x_{5}], x_{43}]=−δ_{6}x_{5};
if also δ_{6} =0 then [[X, x_{6}], x_{51}]=β_{2}x_{18}; if also β_{2} =0 then [[X, x_{1}], x_{51}]=−γ_{5}x_{18};
if also γ_{5} =0 then [[X, x_{6}], x_{18}]=−α_{8}x_{18}; if also α_{8} =0 then [[X, x_{6}], x_{39}]=−ϕ_{1}x_{6};
if also ϕ_{1} =0 then [[X, x_{1}], x_{29}]=γ_{8}x_{18}; if also γ_{8} =0 then [[X, x_{1}], x_{31}]=β_{1}x_{16};
if also β_{1} =0 then [[X, x_{3}], x_{10}]=−δ_{1}x_{18}; if also δ_{1} =0 then [X, x_{27}]=−η_{1}x_{27};
if also η_{1} =0 then [X, x_{28}]=−ϵ_{2}x_{28};  if also ϵ_{2} =0 then [[X, x_{4}], x_{6}]=γ_{4}x_{18};
if also γ_{4} =0 then [[X, x_{1}], x_{33}]=−β_{6}x_{16}; if also β_{6} =0 then [[X, x_{2}], x_{52}]=δ_{7}x_{4};
if also δ_{7} =0 then [[X, x_{1}], x_{35}]=β_{8}x_{16}; if also β_{8} =0 then [[X, x_{1}], x_{6}]=γ_{3}x_{16};
if also γ_{3} =0 then [X, x_{18}]=−ξx_{18};  if also ξ =0 then [X, x_{16}]=α_{4}x_{18};
if also α_{4} =0 then [X, x_{31}]=ψ_{1}x_{27};  if also ψ_{1} =0 then [X, x_{33}]=δ_{3}x_{27};
if also δ_{3} =0 then [X, x_{39}]=α_{3}x_{6};  if also α_{3} =0 then [X, x_{30}]=−ψ_{2}x_{29};
if also ψ_{2} =0 then [X, x_{32}]=α_{5}x_{3};  if also α_{5} =0 then [X, x_{26}]=−α_{2}x_{3};
if also α_{2} =0 then [X, x_{5}]=−β_{7}x_{18};  if also β_{7} =0 then [X, x_{7}]=−γ_{1}x_{15};

if also γ_{1} =0 then X =δ_{4}x_{29}.

Q.E.D.

3.1 Studying the ideal I

We are going to prove that

I =< {x_{i}}_{i=1,...,25} ∪ {x_{26} + x_{34} + x_{37}} >  (20)

is the only proper nonzero ideal of f_{4}(O_{s}, −) in the characteristic two case. In this section we will show that I is a minimal ideal and in the next one we will finish proving that it is also maximal. From Theorem 1, we prove that the ideal I is minimal: the ideal generated by a generic element is the whole algebra or it contains I.

Let us consider the ideal I as a Lie algebra. Is it simple? We can take the basis B_{I} := {y_{i} : i = 1, …, 26} of I, where

y_{i} := x_{i}, i = 1, …, 25 and y_{26} := x_{26} + x_{34} + x_{37}.  (21)

If we take y_{1} and make Lie products with the rest of the elements of B_{I} we obtain, at a first step, the set

{y_{1}, y_{10}, y_{11}, y_{12}, y_{13}, y_{14}, y_{15}, y_{16}, y_{17}, y_{18},

y_{19}, y_{20}, y_{21}, y_{22}, y_{23}, y_{24}, y_{25}}  (22)

where we have eliminated the null elements and added y_{1}. If we make Lie brackets again we obtain all the elements of the basis. That is, the ideal generated by y_{1} is I. In the same way, the ideal generated by y_{i}, i = 2, …, 26, is also I. Let Y be any nonzero element:

Y = ∑_{i=1}^{26} α_{i}y_{i}.  (23)

We only have then to repeat a similar argument as we did to prove that I is minimal to conclude that I is simple as a Lie algebra because in each step we have that the ideal generated by some y_{i}, that is, I, is contained in the one generated by Y. As [[[Y, y_{1}], y_{2}], y_{10}] = α_{11}y_{2}, if α_{11} ≠ 0, we are done. If α_{11} = 0, then proceed as follows:

If α_{11} = 0 then [[Y, y_{2}], y_{10}] = α_{3}y_{10}; if also α_{3} = 0, [[Y, y_{2}], y_{12}] = α_{15}y_{2};
if also α_{15} = 0, [[Y, y_{2}], y_{13}] = α_{16}y_{2}; if also α_{16} = 0, [[Y, y_{2}], y_{14}] = α_{17}y_{2};
if also α_{17} = 0, [[Y, y_{2}], y_{19}] = α_{18}y_{2}; if also α_{18} = 0, [[Y, y_{1}], y_{19}] = α_{10}y_{2};
if also α_{10} = 0, [[Y, y_{2}], y_{7}] = α_{20}y_{19}; if also α_{20} = 0, [[Y, y_{1}], y_{23}] = α_{12}y_{2};
if also α_{12} = 0, [[Y, y_{2}], y_{5}] = α_{22}y_{23}; if also α_{22} = 0, [[Y, y_{1}], y_{2}] = α_{21}y_{13};
if also α_{21} = 0, [Y, y_{2}] = α_{26}y_{2};  if also α_{26} = 0, [[Y, y_{1}], y_{10}] = α_{19}y_{2};
if also α_{19} = 0, [[Y, y_{1}], y_{12}] = α_{23}y_{2}; if also α_{23} = 0, [[Y, y_{1}], y_{13}] = α_{24}y_{2};
if also α_{24} = 0, [[Y, y_{1}], y_{5}] = α_{14}y_{23}; if also α_{14} = 0, [Y, y_{5}] = α_{8}y_{1};
if also α_{8} = 0, [Y, y_{9}] = α_{6}y_{1};  if also α_{6} = 0, [[Y, y_{1}], y_{4}] = α_{13}y_{25};
if also α_{13} = 0, [Y, y_{1}] = α_{25}y_{25};  if also α_{25} = 0, [Y, y_{3}] = α_{2}y_{1};
if also α_{2} = 0, [Y, y_{4}] = α_{7}y_{1};  if also α_{7} = 0, [Y, y_{6}] = α_{9}y_{1};
if also α_{9} = 0, [Y, y_{7}] = α_{4}y_{1};  if also α_{4} = 0, [Y, y_{8}] = α_{5}y_{1};

if also α_{5} = 0, Y = α_{1}y_{1}.

We can stablish then the following theorem where we have included a Cartan decomposition of I:

Theorem 2 I is a 26-dimensional simple Lie algebra with a bidimensional Cartan subalgebra and Cartan decomposition as follows:

I = H ⊕ I_{α} ⊕ I_{β} ⊕ I_{α+β}, (24)
H =< y_{1}, y_{26} >, (25)
I_{α} =< y_{18}, …, y_{25} >, I_{β} =< y_{2}, …, y_{9} >, I_{α+β} =< y_{10}, …, y_{17} >, (26)
α : H → F; α(y_{1}) = 1, α(y_{26}) = 0, (27)
β : H → F; β(y_{1}) = 0, α(y_{26}) = 1, (28)
α + β : H → F; (α + β)(y_{1}) = 1, (α + β)(y_{26}) = 1, (29)
ad(y_{1})|_{I_{α}} = 1_{I_{α}}, ad(y_{1})|_{I_{β}} = 0, ad(y_{1})|_{I_{α+β}} = 1_{I_{α+β}}, (30)
ad(y_{26})|_{I_{α}} = 0, ad(y_{26})|_{I_{β}} = 1_{I_{β}}, ad(y_{26})|_{I_{α+β}} = 1_{I_{α+β}} (31)

[I_{α}, I_{α}]  [I_{β}, I_{β}] ; \begin{aligned}( & \begin{matrix}0 & y_{26} & 0 & 0 & 0 & 0 & 0 & 0 \\ y_{26} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & y_{26} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & y_{26} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & y_{26} \\ 0 & 0 & y_{26} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & y_{26} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & y_{26} & 0 & 0 & 0 \end{matrix} & ) \\ & & ( & \begin{matrix}0 & y_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ y_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & y_{1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & y_{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & y_{1} \\ 0 & 0 & y_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & y_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & y_{1} & 0 & 0 & 0 \end{matrix} & ) \end{aligned} (32)

\begin{aligned}& [I_{α+β}, I_{α+β}] \\ ( & \begin{matrix}0 & y_{1} + y_{26} & 0 & 0 & 0 & 0 & 0 & 0 \\ y_{1} + y_{26} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & y_{1} + y_{26} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & y_{1} + y_{26} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & y_{1} + y_{26} \\ 0 & 0 & y_{1} + y_{26} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & y_{1} + y_{26} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & y_{1} + y_{26} & 0 & 0 & 0 \end{matrix} & ) \end{aligned} (33)

[I_{α}, I_{β}]  [I_{β}, I_{α+β}] ; \begin{aligned}( & \begin{matrix}y_{10} & 0 & 0 & 0 & 0 & y_{15} & y_{16} & y_{17} \\ 0 & y_{11} & y_{12} & y_{13} & y_{14} & 0 & 0 & 0 \\ y_{12} & 0 & 0 & y_{17} & y_{16} & y_{11} & 0 & 0 \\ y_{13} & 0 & y_{17} & 0 & y_{15} & 0 & y_{11} & 0 \\ y_{14} & 0 & y_{16} & y_{15} & 0 & 0 & 0 & y_{11} \\ 0 & y_{15} & y_{10} & 0 & 0 & 0 & y_{14} & y_{13} \\ 0 & y_{16} & 0 & y_{10} & 0 & y_{14} & 0 & y_{12} \\ 0 & y_{17} & 0 & 0 & y_{10} & y_{13} & y_{12} & 0 \end{matrix} & ) \\ & & ( & \begin{matrix}0 & y_{19} & 0 & 0 & 0 & y_{23} & y_{24} & y_{25} \\ y_{18} & 0 & y_{20} & y_{21} & y_{22} & 0 & 0 & 0 \\ 0 & y_{20} & 0 & y_{25} & y_{24} & y_{18} & 0 & 0 \\ 0 & y_{21} & y_{25} & 0 & y_{23} & 0 & y_{18} & 0 \\ 0 & y_{22} & y_{24} & y_{23} & 0 & 0 & 0 & y_{18} \\ y_{23} & 0 & y_{19} & 0 & 0 & 0 & y_{22} & y_{21} \\ y_{24} & 0 & 0 & y_{19} & 0 & y_{22} & 0 & y_{20} \\ y_{25} & 0 & 0 & 0 & y_{19} & y_{21} & y_{20} & 0 \end{matrix} & ) \end{aligned} (34)

\begin{aligned}& [I_{α+β}, I_{α}] \\ ( & \begin{matrix}0 & y_{2} & y_{4} & y_{5} & y_{6} & 0 & 0 & 0 \\ y_{3} & 0 & 0 & 0 & 0 & y_{7} & y_{8} & y_{9} \\ y_{4} & 0 & 0 & y_{9} & y_{8} & y_{2} & 0 & 0 \\ y_{5} & 0 & y_{9} & 0 & y_{7} & 0 & y_{2} & 0 \\ y_{6} & 0 & y_{8} & y_{7} & 0 & 0 & 0 & y_{2} \\ 0 & y_{7} & y_{3} & 0 & 0 & 0 & y_{6} & y_{5} \\ 0 & y_{8} & 0 & y_{3} & 0 & y_{6} & 0 & y_{4} \\ 0 & y_{9} & 0 & 0 & y_{3} & y_{5} & y_{4} & 0 \end{matrix} & ) \end{aligned} (35)

3.2 Studying the quotient f_{4}(O_{s}, −)/I

We are going to prove that the quotient f_{4}(O_{s}, −)/I is also simple, that is, the ideal I is not only minimal, but also maximal: it is unique among the proper nonzero ideals. We can extend the basis B_{I} with y_{i} := x_{i}, i = 27, …, 52, to construct a basis of f_{4}(O_{s}, −). Thus, a basis of f_{4}(O_{s}, −)/I is

B_{c} := \{\overline{y_{i}} = y_{i} + I : i = 27, …, 52\}.  (36)

By taking the projection P onto the subspace generated by {y_{27}, …, y_{52}}, it is easy to define a Lie bracket on f_{4}(O_{s}, −)/I as

< x + I, y + I >:= P ([x, y]) + I, x, y ∈ f_{4}(O_{s}, −).  (37)

Then, let us take \overline{y_{27}}. Performing the products with the rest of the elements of B_{c} we obtain, \overline{y_{27}} included, the linear independent set

\{\overline{y_{27}}, \overline{y_{28}}, \overline{y_{29}}, \overline{y_{31}}, \overline{y_{32}}, \overline{y_{33}}, \overline{y_{34}}, \overline{y_{48}}, \overline{y_{51}}, \overline{y_{52}}\}.  (38)

If we add to this set the one obtained by making again products with the elements in B_{c} we obtain the elements of B_{c} except \overline{y_{45}}, which is included in the next level of depth. Thus, the ideal generated by \overline{y_{27}} in the quotient f_{4}(O_{s}, −)/I is the whole algebra. The same occurs with the rest of the elements of the basis. We repeat the previous argument to prove that any element of the algebra generates the quotient. If Z is a generic element in the quotient

Z = ∑_{i=27}^{52} α_{i}\overline{y_{i}}  (39)

we have that [[[Z, \overline{y_{27}}], \overline{y_{28}}], \overline{y_{33}}] = −α_{45}\overline{y_{27}}. If α_{45} ≠ 0, we are done, but, if α_{45} = 0, then proceed as follows:

If α_{45} =0 then [[Z, \overline{y_{27}}], \overline{y_{28}}] = −α_{46}\overline{y_{27}}; if also α_{46} =0, [[Z, \overline{y_{27}}], \overline{y_{29}}] = −α_{47}\overline{y_{27}};
if also α_{47} =0, [[Z, \overline{y_{27}}], \overline{y_{31}}] = −α_{43}\overline{y_{27}}; if also α_{43} =0, [[Z, \overline{y_{27}}], \overline{y_{30}}] = α_{44}\overline{y_{29}};
if also α_{44} =0, [[Z, \overline{y_{27}}], \overline{y_{33}}] = −α_{38}\overline{y_{27}}; if also α_{38} =0, [[Z, \overline{y_{27}}], \overline{y_{35}}] = α_{41}\overline{y_{31}};
if also α_{41} =0, [[Z, \overline{y_{27}}], \overline{y_{36}}] = α_{39}\overline{y_{31}}; if also α_{39} =0, [[Z, \overline{y_{27}}], \overline{y_{34}}] = α_{40}\overline{y_{31}};
if also α_{40} =0, [Z, \overline{y_{27}}] = −α_{37}\overline{y_{27}};  if also α_{37} =0, [[Z, \overline{y_{28}}], \overline{y_{30}}] = −α_{42}\overline{y_{28}};
if also α_{42} =0, [[Z, \overline{y_{28}}], \overline{y_{33}}] = −α_{34}\overline{y_{27}}; if also α_{34} =0, [[Z, \overline{y_{28}}], \overline{y_{35}}] = α_{36}\overline{y_{31}};
if also α_{36} =0, [[Z, \overline{y_{28}}], \overline{y_{34}}] = α_{35}\overline{y_{29}}; if also α_{35} =0, [Z, \overline{y_{28}}] = −α_{33}\overline{y_{27}};
if also α_{33} =0, [[Z, \overline{y_{29}}], \overline{y_{33}}] = −α_{49}\overline{y_{27}}; if also α_{49} =0, [Z, \overline{y_{29}}] = −α_{48}\overline{y_{27}};
if also α_{48} =0, [[Z, \overline{y_{30}}], \overline{y_{33}}] = −α_{51}\overline{y_{27}}; if also α_{51} =0, [Z, \overline{y_{30}}] = −α_{52}\overline{y_{29}};
if also α_{52} =0, [Z, \overline{y_{31}}] = −α_{50}\overline{y_{29}};  if also α_{50} =0, [[Z, \overline{y_{33}}], \overline{y_{34}}] = α_{30}\overline{y_{31}};
if also α_{30} =0, [Z, \overline{y_{33}}] = α_{28}\overline{y_{27}};  if also α_{28} =0, [Z, \overline{y_{35}}] = −α_{32}\overline{y_{31}};
if also α_{32} =0, [Z, \overline{y_{36}}] = −α_{29}\overline{y_{31}};  if also α_{29} =0, [Z, \overline{y_{34}}] = −α_{31}\overline{y_{31}};
if also α_{31} =0, [Z, \overline{y_{37}}] = α_{27}\overline{y_{27}};  if also α_{27} =0 then Z = 0.

We can affirm then that the quotient is simple and, as a consequence, I is maximal and it is the only proper nonzero ideal in f_{4}(O_{2}, −), if the characteristic is two^{2}.

Theorem 3 Let f_{4}(O_{s}, −) be the Lie algebra of derivations of the quadratic Jordan algebra H_{3}(O_{s}, −) in the split characteristic two case. Then,

i. A generic element of f_{4}(O_{s}, −) is as (55), in the basis (3).

ii. The Lie algebra f_{4}(O_{s}, −) is not simple:

(a) There exists only one proper nonzero ideal, I, in f_{4}(O_{s}, −), see (19), which is 26-dimensional (its structure is given in Theorem 2).

(b) I and the quotient algebra W := f_{4}(O_{s}, −)/I are isomorphic simple Lie algebras.

Proof.

After all, we only have to prove that I and W are isomorphic. For that, we are going to show first that they have a similar Cartan decomposition and secondly we will use this decomposition to define an isomorphism. Using the previous definitions it is easy to check that the quotient has a bidimensional Cartan decomposition:

2If the characteristic is not two, the Lie algebra f_{4}(C, −) is simple (see [7] and [3]).

W = K ⊕ W_{α} ⊕ W_{β} ⊕ W_{α+β}, (40)
K =< \overline{y_{34}}, \overline{y_{37}} >, (41)
W_{α} =< \overline{y_{28}}, \overline{y_{46}}, \overline{y_{48}}, \overline{y_{43}}, \overline{y_{41}}, \overline{y_{39}}, \overline{y_{31}}, \overline{y_{52}} >, (42)
W_{β} =< \overline{y_{35}}, \overline{y_{49}}, \overline{y_{27}}, \overline{y_{50}}, \overline{y_{30}}, \overline{y_{45}}, \overline{y_{36}}, \overline{y_{42}} >, (43)
W_{α+β} =< \overline{y_{29}}, \overline{y_{47}}, \overline{y_{33}}, \overline{y_{44}}, \overline{y_{40}}, \overline{y_{38}}, \overline{y_{32}}, \overline{y_{51}} >, (44)
α : K → F; α(\overline{y_{34}}) = 1, α(\overline{y_{34}}) = 0, (45)
β : K → F; β(\overline{y_{37}}) = 0, α(\overline{y_{37}}) = 1, (46)
α + β : K → F; (α + β)(\overline{y_{34}}) = 1, (α + β)(\overline{y_{37}}) = 1, (47)
ad(\overline{y_{34}})|_{W_{α}} = 1_{W_{α}}, ad(\overline{y_{34}})|_{W_{β}} = 0, ad(\overline{y_{34}})|_{W_{α+β}} = 1_{W_{α+β}}, (48)
ad(\overline{y_{37}})|_{W_{α}} = 0, ad(\overline{y_{37}})|_{W_{β}} = 1_{W_{β}}, ad(\overline{y_{37}})|_{W_{α+β}} = 1_{W_{α+β}} (49)

\begin{aligned}& [W_{α}, W_{α}] \\ ( & \begin{matrix}0 & \overline{y_{37}} & 0 & 0 & 0 & 0 & 0 & 0 \\ \overline{y_{37}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \overline{y_{37}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \overline{y_{37}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \overline{y_{37}} \\ 0 & 0 & \overline{y_{37}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \overline{y_{37}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \overline{y_{37}} & 0 & 0 & 0 \end{matrix} & ) \end{aligned} (50)

\begin{aligned}& [W_{β}, W_{β}] \\ ( & \begin{matrix}0 & \overline{y_{34}} & 0 & 0 & 0 & 0 & 0 & 0 \\ \overline{y_{34}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \overline{y_{34}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \overline{y_{34}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \overline{y_{34}} \\ 0 & 0 & \overline{y_{34}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \overline{y_{34}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \overline{y_{34}} & 0 & 0 & 0 \end{matrix} & ) \end{aligned} (51)

\begin{aligned}& [W_{α+β}, W_{α+β}] \\ ( & \begin{matrix}0 & \overline{y_{34}+y_{37}} & 0 & 0 & 0 & 0 & 0 & 0 \\ \overline{y_{34}+y_{37}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \overline{y_{34}+y_{37}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \overline{y_{34}+y_{37}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \overline{y_{34}+y_{37}} \\ 0 & 0 & \overline{y_{34}+y_{37}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \overline{y_{34}+y_{37}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \overline{y_{34}+y_{37}} & 0 & 0 & 0 \end{matrix} & ) \end{aligned} (52)

[W_{α}, W_{β}]  [W_{β}, W_{α+β}] ; \begin{aligned}( & \begin{matrix}\overline{y_{29}} & 0 & 0 & 0 & 0 & \overline{y_{38}} & \overline{y_{32}} & \overline{y_{51}} \\ 0 & \overline{y_{47}} & \overline{y_{33}} & \overline{y_{44}} & \overline{y_{40}} & 0 & 0 & 0 \\ \overline{y_{33}} & 0 & 0 & \overline{y_{51}} & \overline{y_{32}} & \overline{y_{47}} & 0 & 0 \\ \overline{y_{44}} & 0 & \overline{y_{51}} & 0 & \overline{y_{38}} & 0 & \overline{y_{47}} & 0 \\ \overline{y_{40}} & 0 & \overline{y_{32}} & \overline{y_{38}} & 0 & 0 & 0 & \overline{y_{47}} \\ 0 & \overline{y_{38}} & \overline{y_{29}} & 0 & 0 & 0 & \overline{y_{40}} & \overline{y_{44}} \\ 0 & \overline{y_{32}} & 0 & \overline{y_{29}} & 0 & \overline{y_{40}} & 0 & \overline{y_{33}} \\ 0 & \overline{y_{51}} & 0 & 0 & \overline{y_{29}} & \overline{y_{44}} & \overline{y_{33}} & 0 \end{matrix} & ) \\ & & ( & \begin{matrix}0 & \overline{y_{46}} & 0 & 0 & 0 & \overline{y_{39}} & \overline{y_{31}} & \overline{y_{52}} \\ \overline{y_{28}} & 0 & \overline{y_{48}} & \overline{y_{43}} & \overline{y_{41}} & 0 & 0 & 0 \\ 0 & \overline{y_{48}} & 0 & \overline{y_{52}} & \overline{y_{31}} & \overline{y_{28}} & 0 & 0 \\ 0 & \overline{y_{43}} & \overline{y_{52}} & 0 & \overline{y_{39}} & 0 & \overline{y_{28}} & 0 \\ 0 & \overline{y_{41}} & \overline{y_{31}} & \overline{y_{39}} & 0 & 0 & 0 & \overline{y_{28}} \\ \overline{y_{39}} & 0 & \overline{y_{46}} & 0 & 0 & 0 & \overline{y_{41}} & \overline{y_{43}} \\ \overline{y_{31}} & 0 & 0 & \overline{y_{46}} & 0 & \overline{y_{41}} & 0 & \overline{y_{48}} \\ \overline{y_{52}} & 0 & 0 & 0 & \overline{y_{46}} & \overline{y_{43}} & \overline{y_{48}} & 0 \end{matrix} & ) \end{aligned} (53)

\begin{aligned}& [W_{α+β}, W_{α}] \\ ( & \begin{matrix}0 & \overline{y_{35}} & \overline{y_{27}} & \overline{y_{50}} & \overline{y_{30}} & 0 & 0 & 0 \\ \overline{y_{49}} & 0 & 0 & 0 & 0 & \overline{y_{45}} & \overline{y_{36}} & \overline{y_{42}} \\ \overline{y_{27}} & 0 & 0 & \overline{y_{42}} & \overline{y_{36}} & \overline{y_{35}} & 0 & 0 \\ \overline{y_{50}} & 0 & \overline{y_{42}} & 0 & \overline{y_{45}} & 0 & \overline{y_{35}} & 0 \\ \overline{y_{30}} & 0 & \overline{y_{36}} & \overline{y_{45}} & 0 & 0 & 0 & \overline{y_{35}} \\ 0 & \overline{y_{45}} & \overline{y_{49}} & 0 & 0 & 0 & \overline{y_{30}} & \overline{y_{50}} \\ 0 & \overline{y_{36}} & 0 & \overline{y_{49}} & 0 & \overline{y_{30}} & 0 & \overline{y_{27}} \\ 0 & \overline{y_{42}} & 0 & 0 & \overline{y_{49}} & \overline{y_{50}} & \overline{y_{27}} & 0 \end{matrix} & ) \end{aligned} (54)

The isomorphism, under this structure and using Theorem 2, is now clear: Φ : I ⟶ W such that

\begin{matrix}y_{1} & ↦ & \overline{y_{34}} \\ y_{26} & ↦ & \overline{y_{37}} \end{matrix}  \begin{matrix}y_{18} & ↦ & \overline{y_{28}} \\ y_{19} & ↦ & \overline{y_{46}} \\ y_{20} & ↦ & \overline{y_{48}} \\ y_{21} & ↦ & \overline{y_{43}} \\ y_{22} & ↦ & \overline{y_{41}} \\ y_{23} & ↦ & \overline{y_{39}} \\ y_{24} & ↦ & \overline{y_{31}} \\ y_{25} & ↦ & \overline{y_{52}} \end{matrix}  \begin{matrix}y_{2} & ↦ & \overline{y_{35}} \\ y_{3} & ↦ & \overline{y_{49}} \\ y_{4} & ↦ & \overline{y_{27}} \\ y_{5} & ↦ & \overline{y_{50}} \\ y_{6} & ↦ & \overline{y_{30}} \\ y_{7} & ↦ & \overline{y_{45}} \\ y_{8} & ↦ & \overline{y_{36}} \\ y_{9} & ↦ & \overline{y_{42}} \end{matrix}  \begin{matrix}y_{10} & ↦ & \overline{y_{29}} \\ y_{11} & ↦ & \overline{y_{47}} \\ y_{12} & ↦ & \overline{y_{33}} \\ y_{13} & ↦ & \overline{y_{44}} \\ y_{14} & ↦ & \overline{y_{40}} \\ y_{15} & ↦ & \overline{y_{38}} \\ y_{16} & ↦ & \overline{y_{32}} \\ y_{17} & ↦ & \overline{y_{51}} \end{matrix}

Q.E.D.

References

[1] P. Alberca Bjerregaard, A. Elduque Palomo, C. Martín González and F. J. Navarro Márquez. On the Cartan-Jacobson Theorem. 1999. To appear in Journal of Algebra.

[2] P. Alberca Bjerregaard and C. Martín González. Computing the derivation Lie Algebra of the quadratic Jordan Algebra H_{3}(O_{s}, −) at any characteristic. 2001. Preprint.

[3] N. Jacobson. Structure and Representations of Jordan Algebras. Amer. Math. Soc. Coll. Publ. Vol. 39. Providence. R. I. 1965.

[4] O. Loos. Jordan Pairs. Lecture Notes in Mathematics. vol. 460. Springer-Verlag. Berlin. Heidelberg. New York. 1975.

[5] M. Postnikov. Lie groups and Lie algebras. Edit. Mir, 1985.

[6] L. J. Rylands, D. E. Taylor. Constructions for Octonion and Exceptional Jordan Algebras. Designs, Codes and Cryptography. 21. 191-203. 2000.

[7] R. D. Schafer. An Introduction to Nonassociative Algebras. Academic Press. 1966.

[8] K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov and A. I. Shirshov. Rings that are nearly associative. Academic Press, Inc. N.Y., 1982.

(\begin{matrix}0 & 0 & 0 & α_{1} & α_{2} & α_{3} & α_{4} & α_{5} & α_{6} & α_{7} & α_{8} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & β_{1} & β_{2} & β_{3} & β_{4} & β_{5} & β_{6} & β_{7} & β_{8} \\ 0 & 0 & 0 & −α_{1} & −α_{2} & −α_{3} & −α_{4} & −α_{5} & −α_{6} & −α_{7} & −α_{8} & γ_{1} & γ_{2} & γ_{3} & γ_{4} & γ_{5} & γ_{6} & γ_{7} & γ_{8} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & −γ_{1} & −γ_{2} & −γ_{3} & −γ_{4} & −γ_{5} & −γ_{6} & −γ_{7} & −γ_{8} & −β_{1} & −β_{2} & −β_{3} & −β_{4} & −β_{5} & −β_{6} & −β_{7} & −β_{8} \\ −α_{2} & α_{2} & 0 & δ_{1} & 0 & δ_{2} & δ_{3} & δ_{4} & δ_{5} & δ_{6} & δ_{7} & 0 & β_{2} & 0 & 0 & 0 & β_{6} & β_{7} & β_{8} & γ_{1} & 0 & γ_{3} & γ_{4} & γ_{5} & 0 & 0 & 0 \\ −α_{1} & α_{1} & 0 & 0 & −δ_{1} & ρ_{1} & ρ_{2} & ρ_{3} & ρ_{4} & ρ_{5} & ρ_{6} & β_{1} & 0 & β_{3} & β_{4} & β_{5} & 0 & 0 & 0 & 0 & γ_{2} & 0 & 0 & 0 & γ_{6} & γ_{7} & γ_{8} \\ α_{6} & −α_{6} & 0 & ρ_{4} & δ_{5} & η_{1} & η_{2} & η_{3} & 0 & η_{4} & η_{5} & −β_{6} & 0 & −β_{2} & 0 & 0 & 0 & β_{5} & −β_{4} & γ_{6} & 0 & γ_{2} & 0 & 0 & 0 & −γ_{5} & γ_{4} \\ α_{7} & −α_{7} & 0 & ρ_{5} & δ_{6} & ϵ_{1} & ϵ_{2} & ϵ_{3} & −η_{4} & 0 & ϵ_{4} & −β_{7} & 0 & 0 & −β_{2} & 0 & −β_{5} & 0 & β_{3} & γ_{7} & 0 & 0 & γ_{2} & 0 & γ_{5} & 0 & −γ_{3} \\ α_{8} & −α_{8} & 0 & ρ_{6} & δ_{7} & ϕ_{1} & ϕ_{2} & k_{1} & −η_{5} & −ϵ_{4} & 0 & −β_{8} & 0 & 0 & 0 & −β_{2} & β_{4} & −β_{3} & 0 & γ_{8} & 0 & 0 & 0 & γ_{2} & −γ_{4} & γ_{3} & 0 \\ α_{3} & −α_{3} & 0 & ρ_{1} & δ_{2} & 0 & ψ_{1} & ψ_{2} & −η_{1} & −ϵ_{1} & −ϕ_{1} & 0 & −β_{3} & 0 & −β_{8} & β_{7} & −β_{1} & 0 & 0 & 0 & γ_{3} & 0 & γ_{8} & −γ_{7} & γ_{1} & 0 & 0 \\ α_{4} & −α_{4} & 0 & ρ_{2} & δ_{3} & −ψ_{1} & 0 & χ_{1} & −η_{2} & −ϵ_{2} & −ϕ_{2} & 0 & −β_{4} & β_{8} & 0 & −β_{6} & 0 & −β_{1} & 0 & 0 & γ_{4} & −γ_{8} & 0 & γ_{6} & 0 & γ_{1} & 0 \\ α_{5} & −α_{5} & 0 & ρ_{3} & δ_{4} & −ψ_{2} & −χ_{1} & 0 & −η_{3} & −ϵ_{3} & −k_{1} & 0 & −β_{5} & −β_{7} & β_{6} & 0 & 0 & 0 & −β_{1} & 0 & γ_{5} & γ_{7} & −γ_{6} & 0 & 0 & 0 & γ_{1} \\ 0 & −γ_{2} & γ_{2} & 0 & −β_{2} & −β_{3} & −β_{4} & −β_{5} & 0 & 0 & 0 & ξ & 0 & ϵ_{4} & −η_{5} & η_{4} & −ρ_{4} & −ρ_{5} & −ρ_{6} & −α_{1} & 0 & 0 & 0 & 0 & −α_{6} & −α_{7} & −α_{8} \\ 0 & −γ_{1} & γ_{1} & −β_{1} & 0 & 0 & 0 & 0 & −β_{6} & −β_{7} & −β_{8} & 0 & −ξ & −δ_{2} & −δ_{3} & −δ_{4} & −χ_{1} & ψ_{2} & −ψ_{1} & 0 & −α_{2} & −α_{3} & −α_{4} & −α_{5} & 0 & 0 & 0 \\ 0 & γ_{6} & −γ_{6} & 0 & β_{6} & β_{1} & 0 & 0 & 0 & β_{5} & −β_{4} & −χ_{1} & −ρ_{4} & k_{2} & η_{2} & η_{3} & 0 & −ρ_{3} & ρ_{2} & 0 & −α_{6} & −α_{1} & 0 & 0 & 0 & −α_{5} & α_{4} \\ 0 & γ_{7} & −γ_{7} & 0 & β_{7} & 0 & β_{1} & 0 & −β_{5} & 0 & β_{3} & ψ_{2} & −ρ_{5} & ϵ_{1} & k_{3} & ϵ_{3} & ρ_{3} & 0 & −ρ_{1} & 0 & −α_{7} & 0 & −α_{1} & 0 & α_{5} & 0 & −α_{3} \\ 0 & γ_{8} & −γ_{8} & 0 & β_{8} & 0 & 0 & β_{1} & β_{4} & −β_{3} & 0 & −ψ_{1} & −ρ_{6} & ϕ_{1} & ϕ_{2} & k_{4} & −ρ_{2} & ρ_{1} & 0 & 0 & −α_{8} & 0 & 0 & −α_{1} & −α_{4} & α_{3} & 0 \\ 0 & γ_{3} & −γ_{3} & β_{3} & 0 & 0 & −β_{8} & β_{7} & β_{2} & 0 & 0 & −δ_{2} & ϵ_{4} & 0 & δ_{7} & −δ_{6} & k_{5} & −ϵ_{1} & −ϕ_{1} & −α_{3} & 0 & 0 & α_{8} & −α_{7} & −α_{2} & 0 & 0 \\ 0 & γ_{4} & −γ_{4} & β_{4} & 0 & β_{8} & 0 & −β_{6} & 0 & β_{2} & 0 & −δ_{3} & −η_{5} & −δ_{7} & 0 & δ_{5} & −η_{2} & −k_{3} & −ϕ_{2} & −α_{4} & 0 & −α_{8} & 0 & α_{6} & 0 & −α_{2} & 0 \\ 0 & γ_{5} & −γ_{5} & β_{5} & 0 & −β_{7} & β_{6} & 0 & 0 & 0 & β_{2} & −δ_{4} & η_{4} & δ_{6} & −δ_{5} & 0 & −η_{3} & −ϵ_{3} & −k_{4} & −α_{5} & 0 & α_{7} & −α_{6} & 0 & 0 & 0 & −α_{2} \\ −β_{2} & 0 & β_{2} & −γ_{2} & 0 & γ_{3} & γ_{4} & γ_{5} & 0 & 0 & 0 & α_{2} & 0 & 0 & 0 & 0 & −α_{6} & −α_{7} & −α_{8} & k_{6} & 0 & ϵ_{4} & −η_{5} & η_{4} & δ_{5} & δ_{6} & δ_{7} \\ −β_{1} & 0 & β_{1} & 0 & −γ_{1} & 0 & 0 & 0 & γ_{6} & γ_{7} & γ_{8} & 0 & α_{1} & −α_{3} & −α_{4} & −α_{5} & 0 & 0 & 0 & 0 & −k_{6} & ρ_{1} & ρ_{2} & ρ_{3} & −χ_{1} & ψ_{2} & −ψ_{1} \\ β_{6} & 0 & −β_{6} & γ_{6} & 0 & −γ_{1} & 0 & 0 & 0 & −γ_{5} & γ_{4} & 0 & −α_{6} & α_{2} & 0 & 0 & 0 & −α_{5} & α_{4} & −χ_{1} & δ_{5} & k_{7} & η_{2} & η_{3} & 0 & δ_{4} & −δ_{3} \\ β_{7} & 0 & −β_{7} & γ_{7} & 0 & 0 & −γ_{1} & 0 & γ_{5} & 0 & −γ_{3} & 0 & −α_{7} & 0 & α_{2} & 0 & α_{5} & 0 & −α_{3} & ψ_{2} & δ_{6} & ϵ_{1} & k_{8} & ϵ_{3} & −δ_{4} & 0 & δ_{2} \\ β_{8} & 0 & −β_{8} & γ_{8} & 0 & 0 & 0 & −γ_{1} & −γ_{4} & γ_{3} & 0 & 0 & −α_{8} & 0 & 0 & α_{2} & −α_{4} & α_{3} & 0 & −ψ_{1} & δ_{7} & ϕ_{1} & ϕ_{2} & k_{9} & δ_{3} & −δ_{2} & 0 \\ β_{3} & 0 & −β_{3} & 0 & γ_{3} & 0 & γ_{8} & −γ_{7} & −γ_{2} & 0 & 0 & −α_{3} & 0 & 0 & α_{8} & −α_{7} & α_{1} & 0 & 0 & ρ_{1} & ϵ_{4} & 0 & −ρ_{6} & ρ_{5} & −k_{7} & −ϵ_{1} & −ϕ_{1} \\ β_{4} & 0 & −β_{4} & 0 & γ_{4} & −γ_{8} & 0 & γ_{6} & 0 & −γ_{2} & 0 & −α_{4} & 0 & −α_{8} & 0 & α_{6} & 0 & α_{1} & 0 & ρ_{2} & −η_{5} & ρ_{6} & 0 & −ρ_{4} & −η_{2} & −k_{8} & −ϕ_{2} \\ β_{5} & 0 & −β_{5} & 0 & γ_{5} & γ_{7} & −γ_{6} & 0 & 0 & 0 & −γ_{2} & −α_{5} & 0 & α_{7} & −α_{6} & 0 & 0 & 0 & α_{1} & ρ_{3} & η_{4} & −ρ_{5} & ρ_{4} & 0 & −η_{3} & −ϵ_{3} & −k_{9} \end{matrix} )

where we have defined: k1 := 2ξ + δ1 − ϵ2 − η1, k2 := −ξ − δ1 + η1, k3 := −ξ − δ1 + ϵ2, k4 := ξ − ϵ2 − η1, k5 := ξ + δ1 − η1, k6 := ξ + δ1, k7 := −ξ + η1, k8 := −ξ + ϵ2, k − 9 := ξ + δ1 − ϵ2 − η1,

11