Strongly non-degenerate lie algebras

Let A be a semiprime 2- and 3-torsion free non-commutative associative algebra. We show that the Lie algebra Der(A) of (associative) derivations of A is strongly non-degenerate, which is a strong form of semiprimeness for Lie algebras, under some additional restrictions on the center of A. This result follows from a description of the quadratic annihilator of a general Lie algebra inside appropriate Lie overalgebras. Similar results are obtained for an associative algebra A with involution and the Lie algebra SDer(A) of involution preserving derivations of A. © 2008 American Mathematical Society.