Exercise 6, § 9

Let $\Gamma$ be the 3-dimensional simply connected nilpotent real Lie group defined in Exercise 5 (b) of § 4. Let $\alpha \in \mathbf{R}$ be an irrational number. Let P be the discrete subgroup of $\Gamma \times \mathbf{R}$ consisting of the $((0, 0, x), \alpha x)$, where $x \in \mathbf{Z}$. Let $G = (\Gamma \times \mathbf{R})/P$. Show that $(G, G)$ is not closed in G.