I was studying the following criterion:
Let $k$ be an algebraically closed field, $E$ a $k$-vector space of finite dimension, $B\subset A$ two sub-vector spaces of $\mathscr L(E)$.
Let's define $T:=\{t\in \mathscr L(E),\ \mathrm{ad}_t(A)\subset B\}$, where $\mathrm{ad}_t\in \mathscr L(\mathscr L(E))$ is defined as $\mathrm{ad}_t(s)=ts-st$.
If $t\in T$ is such that for all $u\in T$, $\mathrm{Tr}(tu)=0$, then $t$ is nilpotent.
The question: do you know any application where this criterion is used?