Subspace decompositions invariant under ad/sub e/ for Lie-admissible algebras

Let A be a non-associative algebra of characteristic not two. It is known that the operator ad/sub x/ on A defined by ad/sub x/(y) = [x,y] is a derivation of A+ for all x if and only if A is flexible, and that A is Lie-admissible if and only if ad/sub x/ is a derivative of A- for each x in A. Properties of a decomposition of A into subspaces invariant under ad/sub e/, e an idempotent element, are investigated. The very strong relations between the Peirce decomposition, the Lie-admissible condition and ad/sub e/, e an indempotent, are determined. Multiplicative relations among the subspaces of a are given