Exercise 37, § 9

Let $G$ be a compact real Lie group and $G'$ a finite-dimensional connected real Lie group. Suppose that $L(G)$ is simple. Let $\rho : G \to G'$ be a homomorphism of abstract groups. Suppose that there exists a neighbourhood $V$ of $e_G$ in $G$ such that $\rho(V)$ is relatively compact. Then $\rho$ is continuous. (Let $V'$ be a neighbourhood of $e_{G'}$ in $G'$. Let $V'' \subset V'$ be a neighbourhood of $e_{G'}$ such that

where $n = \dim G$. It can be assumed that $\rho$ is non-trivial. Then $\mathrm{Ker}\, \rho$ is finite by Exercise 25. Therefore $\rho(G)$ is not countable and hence not discrete. Hence there exists $g \in G$ with non-open centralizer such that $\rho(g) \in V''$. Using Exercise 9 of § 4 and its notation, $\rho(M(g, n, V)) \subset V'$ and $M(g, n, V)$ is a neighbourhood of $e_G$ in $G$.)