The notion of admissible
algebra has been introduced by Koecher. They are commutative algebras whose
enveloping Lie algebra (of multiplications) splits into the direct sum of an even and
an odd part. It will be shown here that the class of admissible algebras cannot be
defined by (nonassociative) polynomial identities. This is done by exhibiting
an admissible algebra which possesses a homomorphic image which is not
admissible. The main tool is the relationship between the admissibility of a
homomorphic image of an admissible algebra A, a symmetry property of a
certain ideal of the enveloping Lie algebra of A formed from the kernel of the
homomorphism, and the ideal structure of an algebra constructed by Koecher from
A.
