Corollary 1

Let $g$ be a Lie algebra, $\mathfrak{h}$ a subalgebra reductive in $g$ and $\rho$ a finite-dimensional semi-simple representation of $g$. Then the restriction of $\rho$ to $\mathfrak{h}$ is semi-simple.

It suffices to consider the case where $\rho$ is simple. We adopt the notation V, W of Proposition 4. Let $W_1$ be a subspace of V minimal among the non-zero subspaces stable under $\rho(\mathfrak{h})$. Then $W_1 \subset W$, hence $W \neq \{0\}$ and hence $W = V$.