Exercise 14, § 6

Let $a_1$ be the commutative Lie algebra $\mathbf{R}^4$, identified with $\mathbf{C}^2$. Let $a_2$ be the commutative Lie algebra $\mathbf{R}$. Let $\phi$ be the homomorphism of $a_2$ into $\mathrm{Der}(a_1)$ which maps 1 to the derivation $(z_1, z_2) \mapsto (iz_1, i\sqrt{2}z_2)$. Let a be the semi-direct product of $a_2$ by $a_1$ corresponding to $\phi$.

(a) Show that $(\mathrm{Int}\, a) | a_1$ contains, for all $\phi \in \mathbf{R}$, the automorphism

(b) Show that the closure G of $(\mathrm{Int}\, a) | a_1$ in $\mathbf{GL}(a_1)$ contains, for all $(\phi, \phi') \in \mathbf{R}^2$, the automorphism $(z_1, z_2) \mapsto (e^{i\phi} z_1, e^{i\phi'} z_2)$.

(c) Show that $L(G)$ contains the endomorphism $u : (z_1, z_2) \mapsto (z_1, 0)$ of $a_1$ and that there exists no $x \in a$ such that $u = (\mathrm{ad}\, x) | a_1$.

(d) Deduce that $\mathrm{Int}(a)$ is not closed in $\mathrm{Aut}(a)$.