Exercise 7, § 3

Let $\mathfrak{a}$ be a semi-simple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{a}$, and $\mathfrak{s}$ a semi-simple subalgebra of $\mathfrak{a}$ containing $\mathfrak{h}$. Show that:

$(\mathfrak{a},\mathfrak{h})$ is split $\Leftarrow \Rightarrow (\mathfrak{s},\mathfrak{h})$ is split. Construct an example in which $\mathfrak{a}$ is splittable but $\mathfrak{s}$ is not. (Take $\mathfrak{a}=\mathfrak{s}\mathfrak{p}(4, k)$ and $\mathfrak{s}=\mathfrak{s}\mathfrak{l}(2, k')$, where $k'$ is a quadratic extension of $k.$)