Exercise 12, § 5

Suppose that $K$ is algebraically closed. Let $g$ be a nilpotent Lie $K$-algebra. Let $\rho$ be a finite-dimensional representation of $g$ on a vector space $V$. For every linear form $\lambda$ on $g$, let $V^\lambda$ be the vector subspace of $V$ consisting of the $\xi \in V$ such that, for all $x \in g$, $(\rho(x) - \lambda(x)I)^n \xi = 0$ for sufficiently large $n$.

(a) The $V^\lambda$ are stable under $\rho(g)$ and the sum of the $V^\lambda$ is direct. (Use Exercise 22 of § 4.)

(b) We have $V = \sum V^\lambda$. (If each $\rho(x)$ has a single eigenvalue, $V = V^{\lambda_0}$ by Corollary 2 to Theorem 1. If $\rho(x_0)$ has at least two distinct eigenvalues, $V$ is the direct sum of two non-trivial subspaces which are stable under $\varphi(g)$. Then argue by induction on the dimension of V.)