Exercise 2, § 3

Let $\mathfrak{g}$ be a real Lie algebra of dimension $n, U$ (resp. H) the set of regular elements (resp. of Cartan subalgebras) of $\mathfrak{g}$, and Int($\mathfrak{g}$) the group of inner automorphisms of $\mathfrak{g}$ (Chap. III, §6, no. 2, Def. 2).

a) Show that, if $x$ and $y$ belong to the same connected component of $U,\mathfrak{g}^0(x)$ and $\mathfrak{g}^0(y)$ are conjugate under Int($\mathfrak{g}$). b) Show that the number of connected components of U is finite, and that this number is bounded by a constant $c(n)$ depending only on $n$ (apply Exerc. 2 of App. II).

c) Deduce that the number of orbits of Int($\mathfrak{g}$) on H is $\leq c(n)$.