Ideals in admissible algebras (Q2534206)

A commutative algebra over a field is said to be admissible if the Lie algebra generated by its multiplications is a direct sum of the space spanned by commutator products of an even number of multiplications and the space spanned by commutator products of an odd number of multiplications. This note observes that the class of admissible algebras cannot be given by nonassociative polynomial identities. This is done by exhibiting an admissible Jordan algebra without unit element which possesses a homomorphic image which is not admissible. This is related to a process of Tits and Koecher which constructs Lie algebras out of Jordan algebras.