Let A be a finite-dimensional
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
∑[Rai,Rxi], xi in N, ai in A, where Ra is multiplication by a in A. It was
conjectured in [10] that the various elements xi and ai which occur in the formulation
of σ 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.
