Exercise 9, § 1

Let $\mathfrak{a}$ be a real or complex Lie algebra, and let G be a compact subgroup of Aut($\mathfrak{a}$). Prove that $\mathfrak{a}$ has a Levi subalgebra (Chap. I, §6, no. 8, Def. 7) stable under G (reduce to the case in which the radical of $\mathfrak{a}$ is abelian, and use Integration, Chap. VII, §3, no. 2, Lemma 2).