Exercise 1, § 10

Let $\mathfrak{s}=\mathfrak{s}\mathfrak{l}(2, k)$ and $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(3, k)$. Identify $\mathfrak{s}$ with a subalgebra of $\mathfrak{g}$ by means of an irreducible representation of $\mathfrak{s}$ of degree 3. Show that every subspace of $\mathfrak{g}$ containing $\mathfrak{s}$ and stable under ad$_{\mathfrak{g}}\mathfrak{s}$ is equal to either $\mathfrak{s}$ or $\mathfrak{g}$; deduce that $\mathfrak{s}$ is maximal among the subalgebras of $\mathfrak{g}$ distinct from $\mathfrak{g}$.