Exercise 42, § 9

Let G be a finite-dimensional connected real Lie group. Suppose that

G admits an injective continuous linear representation $\rho : G \to \mathbf{GL}(n, \mathbf{C})$. Then G admits a linear representation $\sigma : G \to \mathbf{GL}(n', \mathbf{C})$ which is a homomorphism of G onto a closed subgroup of $\mathbf{GL}(n', \mathbf{C})$. (By Exercise 41, $\rho((G, G))$ is closed in $\mathbf{GL}(n, \mathbf{C})$ and hence $(G, G)$ is closed in G. Let $\varphi : G \to (G, G)$ be the canonical morphism and $\tau$ an injective linear representation with closed image $G/(G, G)$ which is connected and commutative. Take $\sigma = \rho \oplus (\tau \circ \varphi)$.)