On the Constructions of Tits and Faulkner: an Isomorphism Theorem

Classification theory guarantees the existence of an isomorphism between any two E8’s, at least over an algebraically closed field of characteristic 0. The purpose of this paper is to construct for any Jordan algebra J of degree 3 over a field Φ of characteristic ≠ 2,3 an explicit isomorphism between the algebra obtained from J by Faulkner’s construction and the algebra obtained from the split octonions and J by Tits construction. 2000 Mathematics Subject Classification. 17B60. Let J = J(N,1) be a quadratic Jordan algebra with 1 over a field Φ constructed as in [3] from an admissible nondegenerate cubic form N with base point 1. Let E = {[ α a b β ] |α,β∈ Φ, a,b ∈ J } . (1) Then it is shown in [1] that E is a ternary algebra. Moreover, it is shown in [1] that S = S(E,R(E)) = E⊕ Ẽ⊕Φu⊕Φũ⊕ΦU ⊕D is a Lie algebra of dimension 248, where Ẽ denotes a second copy of E. (Recall from [1] that D is the algebra of derivations of E and R(E) = ΦU ⊕D, the span of {R(x,y) | x,y ∈ E} ∪ {U}. For R = E ⊕Φu, U(E)= {A∈HomΦ(R,R) |uA∈ Φu and EA⊆ E} is a Lie algebra with [AB]=AB−BA. Here U ∈ U(E) such that uU = 2u and xU = x, for all x ∈ E. So, U is in the center of U(E). Also R(x,y)∈U(E), for x,y ∈ E, is defined by uR(x,y)= 〈x,y〉u, zR(x,y)= 〈z,x,y〉, for all z ∈ E, where 〈 ,〉 is the bilinear form and 〈, ,〉 is the ternary form defining a ternary Φ-module, as in [1].) On the other hand, let C be the algebra of split octonions and J an arbitrary Jordan algebra of degree 3. Then according to Tits in [4], L=Der(C)⊕C0⊗J0⊕Der(J) is a Lie algebra, where Der(C) is the algebra of derivations of C and C0 is the subspace of C whose elements have trace 0. Theorem 1. There is an explicit isomorphism between the Lie algebras constructed by Tits and the Faulkner processes. The proof of this theorem is the goal of this paper and the proof completes this paper. Our attempt is to construct an isomorphism Φ : L→ S by breaking Φ into three parts Φ1 restricted to Der(C), Φ2 restricted to C0⊗J0, and Φ3 restricted to Der(J). First we construct a candidate S1 for the image of Φ1 in S which in turn plays a significant role throughout the paper. This S1 is obtained by replacing J by Φ1 in the