Exercise 10, § 4

Let $\mathscr{D}$ be the Lie algebra of derivations of the algebra $g_4$ (Exercise 9). Show that $\mathscr{D}$ is of dimension 7 and of zero centre and that the ideal $\mathscr{D}' \subset \mathscr{D}$ of inner derivations has no supplementary subalgebra in $\mathscr{D}$.