Exercise 12, § 3

a) Let $\mathfrak{g}$ be a solvable Lie algebra, and G a finite subgroup (resp. compact subgroup if $k=\mathbf{R}$ or $\mathbf{C}$) of Aut($\mathfrak{g}$). Show that there exists a Cartan subalgebra of $\mathfrak{g}$ stable under G. (Argue by induction on dim $\mathfrak{g}$, and reduce to

$^7$ For more details, cf. J. DIXMIER, Sous-algèbres de Cartan et décompositions de Levi dans les algèbres de Lie, Trans. Royal Soc. Canada, Vol. L (1956), pp. 17-21. the case in which $\mathfrak{g}$ is an extension of a nilpotent algebra $\mathfrak{g}/\mathfrak{n}$ by a commutative ideal $\mathfrak{n}$ which is a non-trivial simple $\mathfrak{g}/\mathfrak{n}$-module (cf. proof of Th. 3). The Cartan subalgebras of $\mathfrak{g}$ then form an affine space attached to $\mathfrak{n}$, on which G operates. Conclude by an argument with the barycentre.)

b) Let $\mathfrak{g}$ be a solvable Lie algebra and $\mathfrak{s}$ a Lie subalgebra of Der($\mathfrak{g}$). Assume that the $\mathfrak{s}$-module $\mathfrak{g}$ is semi-simple. Show that there exists a Cartan subalgebra of $\mathfrak{g}$ stable under $\mathfrak{s}$. (Same method.)