On Lie-Admissible Algebras Whose Commutator Lie Algebras Are Lie Subalgebras of Prime Associative Algebras

We describe third power associative multiplications * on noncentral Lie ideals of prime algebras and skew elements of prime algebras with involution provided that x * y − y * x = [x, y] for all x, y and the prime algebras in question do not satisfy polynomial identities of low degree. We also obtain necessary and sufficient conditions for these multiplications to be fourth power-associative or flexible.