Here is a result, which I think is interesting in this context (for reference see books on Lie algebras):
Theorem: Let $V$ be a finite-dimensional vector space over a field $K$ of characteristic zero. Let $E\subseteq F$ be two linear subspaces of $End(V)$ and $M=\lbrace A \in End(V) \mid AF-FA \subseteq E\rbrace $. Assume that $tr (AB)=0$ for all $A\in M$. Then $B$ is nilpotent with $tr(B)=tr(B^2)=\cdots = tr(B^n)=0$ for all $n$.