Exercise 4, § 2

Suppose that $\mathbf{K}$ is a field. (a) Verify that there exists a 3-dimensional Lie algebra $g$ with basis $(x, y, z)$ such that $[x, y] = z, [x, z] = [y, z] = 0$. Let $U$ be the enveloping algebra of $g$. Show that the centre of $U$ is the subalgebra generated by 1 and $z$. (Consider, for all $\xi \in g$, the derivation of the symmetric algebra $S$ of $g$ which extends $\mathrm{ad}_g \xi$ and look for the elements of $S$ annihilated by these derivations; then apply the final remark of no. 8.) (b) Verify that there exists a 3-dimensional Lie algebra $g$ with basis $(x, y, z)$ such that $[x, y] = y, [x, z] = z, [y, z] = 0$. Show that the centre of the enveloping algebra of $g$ reduces to the scalars. (Same method.)