Clifford elements in Lie algebras

Abstract

Let L be a Lie algebra over a field F of characteristic zero or p > 3 . An element c ∈ L is called Clifford if adc^3 = 0 and its associated Jordan algebra Lc is the Jordan algebra F ⊕ X defined by a symmetric bilinear form on a vector space X over F . In this paper we prove the following result: Let R be a centrally closed prime ring R of characteristic zero or p > 3 with involution ∗ and let c ∈ Skew(R, ∗) be such that c^3 = 0 , c^2 != 0 and c^2kc = ckc^2 for all k ∈ Skew(R, ∗) . Then c is a Clifford element of the Lie algebra Skew(R, ∗) .