Exercise 22, § 4

(a) Let $u, v$ be two endomorphisms of a vector space $E$ of finite dimension over an algebraically closed field $K$. For every eigenvalue $\lambda$ of $u$ let $E_\lambda$ be the subspace of $E$ consisting of the vectors annihilated by a power of $u - \lambda I$. Show that, if $(\mathrm{ad}\, u)^n v = 0$, the subspaces $E_\lambda$ are stable under $v$. (b) Deduce from (a) that, if $g$ is a nilpotent Lie algebra of endomorphisms of $E$, $E$ is the direct sum of subspaces $F_j$ ($1 \leq j \leq m$) which are stable under $g$ and such that on each $F_j$ the restriction of every element $u \in g$ can be written $\lambda_j(u) I + u_j$, where $\lambda_j(u) \in K$ and $u_j$ is nilpotent. (c) If $K$ is of characteristic 2, $E = K^2$ and $g$ is nilpotent algebra $\mathfrak{sl}(2, K)$, show that $m = 1$ and that possibly $\lambda(u + v) \neq \lambda(u) + \lambda(v)$ for two elements $u, v$ of $g$. (For a sequel to this exercise, cf. § 5, Exercise 12.)