Exercise 22, § 9

Let G be a finite-dimensional connected real Lie group such that $L(G)$ is reductive. Then $G = (V \times S)/N$ where V is a finite-dimensional real vector space, where S is a simply connected semi-simple real Lie group and N is a central discrete subgroup of $V \times S$. Then $G/\overline{D^1}G$ is isomorphic to $V/\mathrm{pr}_1 N$. Hence $G/\overline{D^1}N$ is compact if and only if $\mathrm{pr}_1 N$ generates the vector space V.