Exercise 16, § 3

Let V be a finite dimensional $k$-vector space, and $\mathfrak{g}$ a Lie subalgebra of $\mathfrak{g}\mathfrak{l}(V)$. We are going to show that the following three properties are equivalent: (i) The Cartan subalgebras of $\mathfrak{g}$ are commutative and consist of semi-simple elements.

(ii) Every regular element of $\mathfrak{g}$ is semi-simple.

(iii) The semi-simple elements of $\mathfrak{g}$ are dense in $\mathfrak{g}$ in the Zariski topology. a) Show that (i) $=\Rightarrow$ (ii) $=\Rightarrow$ (iii).

b) Let A be the set of semi-simple elements of $\mathfrak{g}$. Show that A is stable under Aut$_e(\mathfrak{g})$. c) Show that (iii) $=\Rightarrow$ (i). (Reduce (App. I, Exerc. 1) to the case where $k$ is algebraically closed. If $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, show, by using Exerc. 15, that $A\cap \mathfrak{h}$ is dense in $\mathfrak{h}$; since $[x, y] = 0$ if $x\in A\cap \mathfrak{h}, y\in \mathfrak{h}$, deduce that $\mathfrak{h}$ is commutative, from which (i) is immediate.)

d) Assume that $k$ is $\mathbf{R},\mathbf{C}$ or a non-discrete complete ultrametric field of characteristic zero. Give $\mathfrak{g}$ the topology defined by that of $k$. Show that properties (i), (ii), (iii) are equivalent to the following:

(iv) The semi-simple elements of $\mathfrak{g}$ are dense in $\mathfrak{g}$.

(Show that (iv) $=\Rightarrow$ (iii) and (ii) $=\Rightarrow$ (iv), cf. App. I, Exerc. 4.)