Exercise 13, § 3

Let $\mathfrak{g}$ be a Lie algebra and G a finite subgroup of Aut($\mathfrak{g}$). Assume that G is hyper-solvable (Algebra, Chap. I, §6, Exerc. 26). Show that there exists a Cartan subalgebra of $\mathfrak{g}$ stable under G.

(Argue by induction on dim$\mathfrak{g}$. Using Exerc. 12, reduce to the case where $\mathfrak{g}$ is semi-simple. If $G\not=\{1\}$, choose a normal subgroup C of G that is cyclic of prime order (Algebra, loc. cit.). The subalgebra $\mathfrak{s}$ consisting of the elements invariant under C is reductive in $\mathfrak{g}($§1, no. 5), and distinct from $\mathfrak{g}$. By the induction hypothesis, $\mathfrak{s}$ has a Cartan subalgebra $\mathfrak{a}$ stable under G. We have $\mathfrak{s}\not= 0$ (Chap. I, §4, Exerc. $21c$)), so $\mathfrak{a}\not= 0$. The commutant $\mathfrak{z}$ of $\mathfrak{a}$ in $\mathfrak{g}$ is distinct from $\mathfrak{g}$ and stable under G; choose a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{z}$ stable under G, and show that $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, cf. the Cor. to Prop. 3.)

Construct a finite group of automorphisms of $\mathfrak{s}\mathfrak{l}(2,\mathbf{C})$ which is isomorphic to $\mathfrak{A}_4$ (and hence solvable) and which does not leave any Cartan subalgebra stable.