Proposition 7

Let $g$ be a Lie algebra, $h$ a subalgebra reductive in $g$, $\rho$ a representation of $g$ on a vector space $V$ and $W$ the sum of the finite-dimensional subspaces of $V$ which are simple $h$-modules. Then $W$ is stable under $\rho(g)$.

Let $W_0$ be a finite-dimensional simple sub-$h$-module of $V$. We need to prove that $\rho(x)(W_0) \subset W$ for all $x \in g$. Let $M$ denote the vector space $g$ considered as an $h$-module by means of the representation $x \mapsto \operatorname{ad}_g x$ of $h$ on $g$. Then $M \otimes_K W_0$ is a semi-simple $h$-module (Corollary 1 to Theorem 4). Let $\theta$ be the K-linear mapping of $M \otimes_K W_0$ into V defined by $\theta(x \otimes w) = \rho(x)w$. This is an $\mathfrak{h}$-module homomorphism, for if $y \in \mathfrak{h}$ then:

Hence $\theta(M \otimes_K W_0)$ is a finite-dimensional semi-simple $\mathfrak{h}$-module. Hence $\theta(M \otimes_K W_0) \subset W$, that is $\rho(x)(W_0) \subset W$ for all $x \in g$.