Exercise 7, § 2

Let $\mathfrak{s}$ be a Lie algebra, $\mathfrak{a}$ a Cartan subalgebra of $\mathfrak{s}$ and V an $\mathfrak{s}$-module. Let $\mathfrak{g}=\mathfrak{s}\times V$ be the semi-direct product of $\mathfrak{s}$ by V. Show that $\mathfrak{a}\times V^0(\mathfrak{a})$ is a Cartan subalgebra of $\mathfrak{g}$.