Corollary 3

Let G be a finite-dimensional connected Lie group and H a normal connected Lie subgroup of G. The canonical morphism of $\pi_1(H)$ into $\pi_1(G)$ is injective.

Let $G_1$ be the universal covering of G and $\lambda$ the canonical mapping of $G_1$ onto G. The Lie algebra of $G_1$ is identified with $L(G)$. The Lie subgroup $\lambda^{-1}(H)$ of $G_1$ is normal in $G_1$ and its Lie algebra is $L(H)$. Let $H_1$ be the identity component of $\lambda^{-1}(H)$ and $\lambda_1 = \lambda | H_1$. Then $L(H_1) = L(H)$ and hence $\lambda_1$ is an étale morphism of $H_1$ onto H. On the other hand, $H_1$ is simply connected (Corollary 2) and hence is identified with the universal covering of H. Then, by General Topology, Chapter XI, the canonical morphism of $\pi_1(H)$ into $\pi_1(G)$ is identified with the canonical injection of Ker $\lambda_1$ into Ker $\lambda$.