Exercise 2, § 1

Assume that $k$ is perfect. Let V be a finite dimensional vector space, and $u, v\in$ End(V). Let $u_s, u_n, v_s, v_n$ be the semi-simple and nilpotent components of $u, v$. The following conditions are equivalent: (i) there exists an integer $m$ such that (ad $u$)$^mv= 0$; (ii) $u_s$ and $v$ commute. (To prove (i) $=\Rightarrow$ (ii), reduce to the case where $k$ is algebraically closed and use Lemma 1 (ii).)