Lemma 2

Let $g$ be a Lie algebra. Then the following two conditions are equivalent: (a) All finite-dimensional linear representations of $g$ are semi-simple. (b) Given a linear representation $\varphi$ of $g$ on a finite-dimensional vector space $V$ and a vector subspace W of codimension 1 such that $\rho(x)(V) \subset W$ for all $x \in g$, there exists a supplementary line of W which is stable under $\rho(g)$ (and hence annihilated by $\rho(g)$).

Clearly (a) implies (b). Suppose that (b) holds. Let $\sigma$ be a finite-dimensional representation of $g$ on a vector space M and N a vector subspace which is stable under $\sigma(g)$. Let $\mu$ be the representation of $g$ on $\mathscr{L}(M)$ canonically derived from $\sigma$ (§ 3, no. 3): recall that $\mu(x) = \mathrm{ad}_{\mathscr{L}(M)} \rho(x)$. Let V (resp. W) be the subspace of $\mathscr{L}(M)$ consisting of the linear mappings of M into N whose restriction to N is a homothety (resp. zero); then W is of codimension 1 in V and $\mu(x)(V) \subset W$ for all $x \in g$. By condition (b), there exists $u \in V$ which is annihilated by $\mu(x)$ for all $x \in g$ and whose restriction to N is a non-zero homothety. By multiplying $u$ by a suitable scalar, it can be assumed that $u$ is a projector of M onto N. To say that $\mu(x) \cdot u = 0$ means that $u$ is permutable with $\sigma(x)$. Hence the kernel of $u$ is a supplement of N in M which is stable under $\sigma(x)$ for all $x \in g$. Hence $\sigma$ is semi-simple.