The derived superalgebra of skew elements of a semiprime superalgebra with superinvolution

In this paper we investigate the Lie structure of the derived Lie superalgebra [K,K][K,K], with K the set of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U   is a Lie ideal of [K,K][K,K], then either there exists an ideal J of A   such that the Lie ideal [J∩K,K][J∩K,K] is nonzero and contained in U, or A   is a subdirect sum of A′A′, A″A″, where the image of U   in A′A′ is central, and A″A″ is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center.