Exercise 1, § 4

Let G be a Lie group, H a normal Lie quasi-subgroup of G and $\pi$ the canonical mapping of G onto G/H. There exists on G/H one and only one Lie group structure with the following property: for a homomorphism $\theta$ of G/H into a Lie group G' to be a Lie group morphism, it is necessary and sufficient that $\theta \circ \pi$ be a Lie group morphism. Moreover, L(G/H) is canonically isomorphic to L(G)/L(H). (Let Q be a Lie group germ such that $L(Q) = L(G)/L(H)$. Show that, by shrinking Q if necessary, Q can be identified with an open neighbourhood of 0 in G/H and that Proposition 18 of § 1 can then be applied to G/H.)