Exercise 41, § 9

Let G be a finite-dimensional connected real Lie group and $\varphi : G \to \mathbf{GL}(n, \mathbf{C})$ a continuous homomorphism with finite kernel. Then $\varphi((G, G))$ is closed in $\mathbf{GL}(n, \mathbf{C})$. (Using Exercises 7 (b) and 40 (a), reduce it to the case where G is the semi-direct product of a semi-simple group S and its radical R. Every element of $\varphi((G, R))$ is unipotent and hence $\varphi((G, R))$ is closed. The vector subspace $V_1$ of fixed points of $\varphi((G, R))$ is $\neq \{0\}$. It is stable under $\varphi(G)$. By considering $\mathbf{C}^n/V_1$ arguing by induction on n and using the complete reducibility of $\varphi(S)$, it is seen that it is possible to write

where each $V_i$ is stable under $\varphi(S)$ and $\varphi((G, R))(V_i) \subset V_1 \oplus \cdots \oplus V_{i-1}$ for for all i. On the other hand, $\varphi(S)$ is closed (Exercise 40 (b)). Finally, $\varphi((G, G)) = \varphi(S) \cdot \varphi((G, R))$.)