Exercise 3, § 6

Let $G_1$ be the simply connected nilpotent real Lie group defined in Exercise 5 of § 4. Let $Z$ be the centre of $G_1$, $N$ a non-trivial discrete subgroup of $Z$ and $G = G_1/N$. If $\rho$ is a finite-dimensional analytic linear representation of $G$, $\rho(Z/N)$ is a semi-simple set of automorphisms, for $Z/N$ is compact; but $Z = (G, G)$ and hence $\rho(Z/N)$ is unipotent (Chapter I, § 6, Proposition 6). Therefore $\rho$ is trivial on $Z/N$.