Exercise 9, § 3

Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$ and $x$ an element of $\mathfrak{h}$. Let $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{g}^+$ be the Fitting decomposition (§1, no. 1) of $\mathfrak{g}$ with respect to $\mathfrak{h}$. a) Let $\mathfrak{n}$ be the largest semi-simple $\mathfrak{h}$-submodule contained in $\mathfrak{g}^0(x)\cap \mathfrak{g}^+$. Show that $\mathfrak{n}= 0$ if and only if $\mathfrak{g}^0(x) =\mathfrak{h}$, i.e. $x$ is regular in $\mathfrak{g}$.

b) Show that $\mathfrak{h}\oplus \mathfrak{n}$ is a subalgebra of $\mathfrak{g}$. If $\mathfrak{h}'$ is the intersection of $\mathfrak{h}$ with the commutant of $\mathfrak{n}$, show that $\mathfrak{h}'$ is an ideal of $\mathfrak{h}\oplus \mathfrak{n}$ which contains $\mathscr{D}\mathfrak{h}$ and $x$. Conclude (Exerc. 8) that $\mathfrak{h}'\subset \mathscr{C}_{\infty}(\mathfrak{h}\oplus \mathfrak{n})$, and hence that $x\in \mathscr{C}_{\infty}(\mathfrak{h}\oplus \mathfrak{n})$ and that $x$ belongs to every Cartan subalgebra of $\mathfrak{h}\oplus \mathfrak{n}$.

c) If $\mathfrak{n}\not= 0,\mathfrak{h}\oplus \mathfrak{n}$ is not nilpotent and has infinitely-many Cartan subalgebras (§2, Exerc. 10). Conclude that $x$ belongs to infinitely-many Cartan subalgebras of $\mathfrak{g}$.

d) An element of $\mathfrak{g}$ is regular if and only if it belongs to a unique Cartan subalgebra.