Exercise 4, § 5

Let $\mathfrak{g}$ be a decomposable Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$. Let $\mathfrak{q}$ be a Lie subalgebra of $\mathfrak{g}$ whose identity representation is semi-simple. There exists $a\in$ Aut$_e(\mathfrak{g})$ such that $a(\mathfrak{q})$ is contained in the subalgebra $\mathfrak{m}$ of Prop. 7. (Imitate the proof of Prop. 7 (ii).)