Exercise 8, § 5

Let $\mathfrak{g}$ be a Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$ and $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$.

a) Show that $e(\mathfrak{g}) =e(\mathfrak{h}) +\mathscr{D}\mathfrak{g}=e(\mathfrak{h}) +\mathfrak{g}$.

(Remark that $e(\mathfrak{h})+\mathscr{D}\mathfrak{g}$ is decomposable (Cor. 1 of Th. 1), contains $\mathfrak{g}=\mathfrak{h}+\mathscr{D}\mathfrak{g}$, and is contained in $e(\mathfrak{g})$; thus, it is $e(\mathfrak{g}).$)

b) We have $e(\mathfrak{h})\cap \mathfrak{g}=\mathfrak{h}$ (remark that $e(\mathfrak{h})\cap \mathfrak{g}$ is nilpotent).

c) Let $x$ be an element of the normalizer of $e(\mathfrak{h})$ in $e(\mathfrak{g})$. Show that $x\in e(\mathfrak{h})$. (Write $x=y+z$ with $y\in e(\mathfrak{h}), z\in \mathfrak{g}$, cf. $a$); remark that $[z,\mathfrak{h}]\subset e(\mathfrak{h})\cap \mathfrak{g}=\mathfrak{h}$, hence $z\in \mathfrak{h}.)$

d) Show that $e(\mathfrak{h})$ is a Cartan subalgebra of $e(\mathfrak{g})$.