Exercise 1, § 3

Let $\mathfrak{g}$ be a Lie algebra, $\mathfrak{g}'$ a Cartan subalgebra of $\mathfrak{g}$. Then the conditions of Prop. 3 are satisfied. But an element of $\mathfrak{g}'$, even if it is regular in $\mathfrak{g}'$, is not necessarily regular in $\mathfrak{g}$.