Proposition 33

Let G be a finite-dimensional complex Lie group whose number of connected components is finite, $\varphi$ a finite-dimensional analytic linear representation of G and $G'$ an integral subgroup of the real Lie group G such that $L(G')$ generates $L(G)$ over $\mathbf{C}$. Then, for $\varphi$ to be semi-simple, it is necessary and sufficient that $\varphi|G'$ be semi-simple.

Let $\varphi' = \varphi|G'$. For $\varphi$ (resp. $\varphi'$) to be semi-simple, it is necessary and sufficient that $L(\varphi)$ (resp. $L(\varphi')$) be semi-simple (Theorem 1). Let V be the space of $\varphi$. For a vector subspace of V to be stable under $L(\varphi)(L(G))$, it is necessary and sufficient that it be stable under $L(\varphi')(L(G'))$. Hence the proposition.