Exercise 8, § 5

(a) Let $g_0$ be a 3-dimensional solvable real Lie algebra such that $\mathscr{L}g_0$ is commutative and 2-dimensional. Let $g$ be the algebra derived from $g_0$ by extending the base field from $\mathbf{R}$ to $\mathbf{C}$. For $x \in g$, let $u_x$ be the restriction of $\mathrm{ad}_g x$ to $\mathscr{D}g$. Show that the eigenvalues of $u_x$ either have the same absolute value or are linearly dependent over $\mathbf{R}$. (We have $x = \lambda y + z$ with $z \in \mathscr{D}g$, $y \in g_0$, $\lambda \in \mathbf{C}$ and hence $u_x = \lambda u_y$. Now $u_y$ is the $\mathbf{C}$-linear extension to $\mathscr{D}g$ of an $\mathbf{R}$-linear endomorphism of $\mathscr{D}g_0$.)

(b) Show that there exists a 3-dimensional solvable complex Lie algebra $g$, with $\mathscr{D}g$ commutative and 2-dimensional, and an element $x$ of $g$ such that the restriction of $\mathrm{ad}_g x$ to $\mathscr{D}g$ has eigenvalues which neither have the same absolute value nor are linearly dependent over $\mathbf{R}$. (Construct $g$ as a semi-direct product of a 1-dimensional algebra and a 2-dimensional commutative algebra.)

(c) Show that the algebra constructed in (b) cannot be derived from a real Lie algebra by extending the base field from $\mathbf{R}$ to $\mathbf{C}$.