Corollary 1

Let G and H be Lie groups, $\phi$ a Lie group morphism of G into H, N the kernel of $\phi$ and $h = L(\phi)$. Suppose that G is connected and that H is finite-dimensional. (i) N is a Lie subgroup of G and $L(N) = \mathrm{Ker}\ h$. (ii) Let H' be the integral subgroup of H with Lie algebra Im h. Then $\phi(G) = H'$. (iii) The mapping of G/N into H' derived from $\phi$ when passing to the quotient is a Lie group isomorphism.

(i) has already been proved (§ 3, no. 8, Proposition 28).

Let $\psi$ be the Lie group morphism of G/N into H derived from $\phi$ when passing to the quotient; it is an immersion (§ 3, no. 8, Proposition 28). By Proposition 1, $\psi$ is a Lie group morphism of G/N into H'. This morphism is étale and hence $\psi(G/N) = H'$ since H' is connected; this proves (ii). Then $\psi : G/N \to H'$ is bijective and is a Lie group isomorphism, which proves (iii).