A counter-example to a fixed point conjecture

Let A be a finite-dimentional commutative Jordan algebra over a field F of characteristic zero.Then we may write A -S + N, S a semisimple subalgebra (Wedderburn factor), N the radical of A, [5], [6].If G is a completely reducible group of automorphisms of A, then we may choose S to be invariant under G, [4], If G is finite, then we showed in [10] that any two such G-invariant S were conjugate via an automorphism σ of A which centralizes G and which is a product of exponentials of nilpotent inner derivations of A of the form Σ[i? α{ , R Xi ] 9 Xi in JV, (Li in A, where R a is multiplication by a in A. It was conjectured in [10] that the various elements Xi and di which occur in the formulation of a could be chosen as fixed points of G.This conjecture was based on analogous fixed point results proved for associative and Lie algebras, [7], [8], [9], However, this conjecture is false, and we present in this note a simple counter-example.We consider three-by-three matrices over F. Denoting by e iά the usual matrix units, set e -e n + e 22 , f = e 33 and x = β 31 .Consider the Jordan algebra A with basis e, f, x and multiplication table