Exercise 9, § 4

(a) Verify that the following multiplication tables define two nilpotent Lie algebras $g_3, g_4$ of dimensions 3 and 4:

(b) Show that $g_3$ is isomorphic to $\mathfrak{n}(3, K)$ and also to $\mathfrak{sl}(2, K)$ if $K$ is of characteristic 2.

(c) Let $g_1$ be the unique Lie algebra of dimension 1. Show that the nilpotent Lie algebras of dimension $\leq 4$ are given by the following table:

(Use Exercise 7 (c). If $g$ is nilpotent and 3-dimensional and $\dim \mathscr{D} g = 1$, observe that $\mathscr{D} g$ is contained in the centre of $g$ (Exercise 3 (a)), whence $g = g_3$. If $\dim g = 4$, $\dim \mathscr{D}g = 1$, observe that the bracket on $g$ defines an alternating bilinear form on $g / \mathscr{D}g$, which is necessarily degenerate and hence that there exists a subalgebra $h$ of $g$ contained in the centre of $g$ with $\dim h = 1$, $h \cap \mathscr{D}g = \{0\}$, whence $g = h \times g_3$. If $\dim g = 4$, $\dim \mathscr{D}g = 2$, $\mathscr{D}g$ is commutative; applying Theorem 1 to the restrictions to $\mathscr{D}g$ of the $\mathrm{ad}_g x$ ($x \in g$), show that there exists a commutative ideal $h$ of $g$ with $h \supset \mathscr{D}g$, $\dim h = 3$; let $x \in g$, $x \notin h$; choose a basis of $h$ such that the restriction of $\mathrm{ad}_g x$ to $h$ has a Jordan matrix with respect to this basis; then $g = g_4$.