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 $\mathcal{D}\mathrm{er}(A)$ 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 $\mathrm{SDer}(A)$ of involution preserving derivations of $A$.