Exercise 15, § 3

Assume that $k$ is algebraically closed. Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$, and A a subset of $\mathfrak{g}$. Assume that A is dense in $\mathfrak{g}$ (in the Zariski topology) and stable under Aut$_e(\mathfrak{g})$. Show that $A\cap \mathfrak{h}$ is dense in $\mathfrak{h}$. (Let X be the closure of $A\cap \mathfrak{h}$, and $U =\mathfrak{h}$-- X. Assume that $U\not=\emptyset$. With the notations in Lemma 2, the image under F of $U\times \mathfrak{g}^{\lambda_1}(\mathfrak{h})\times \cdots \times \mathfrak{g}^{\lambda_p}(\mathfrak{h})$ contains a non-empty open subset of $\mathfrak{g}$. Since this image is contained in $\mathfrak{g}$-- A, this contradicts the fact that A is dense in $\mathfrak{g}.$)