Let $k$ be a field and $n$ a nonnegative integer. For any matrix $U\in\mathrm{M}_n\left(k\right)$, let $\mathrm{ad} U$ denote the map $\mathrm{M}_n\left(k\right)\to \mathrm{M}_n\left(k\right),\ V\mapsto UV-VU$. Thus, $\mathrm{ad} U$ is an element of the $k$-algebra $\mathrm{End}_k\left(\mathrm{M}_n\left(k\right)\right)$.

Is it true that for every $n\times n$-matrix $A$ over $k$, and for every $m\in\mathbb N$, the endomorphism $\mathrm{ad}\left(A^m\right)$ can be written in the form $P\left(\mathrm{ad}A\right)$ for some polynomial $P\in k\left[X\right]$ satisfying $P\left(0\right)=0$ ?

I know that this holds when $A$ is diagonalizable, and in that case it is used in the proof of Cartan's Lemma from Lie algebra theory. If it holds generally and can be proven neatly, it could be used to tidy up the proof of Cartan's Lemma (which, in the form I know it, is rather ugly, requiring an algebraic extension of the ground field and the use of Jordan's normal form).