Exercise 26, § 9

Let G be a finite-dimensional connected real Lie group. Suppose that G admits a finite-dimensional injective continuous linear representation $\rho$. Then (G, G) is closed in G. (Let R be the radical of G and S a maximal semi-simple integral subgroup of G. Using Exercise 7 (b), reduce it to the case where G is the semi-direct product of S and R. By Chapter I, § 6, Proposition 6, $\rho$ is unipotent on (G, R). Hence $\rho((G, R))$ is closed in the linear group and therefore (G, R) is closed in G.)