Proposition 31

Let G be a finite-dimensional Lie group with countable base, H a normal Lie subgroup of G and A a Lie subgroup of G. Suppose that K is of characteristic 0 and that AH is closed. Let $\phi$ be the canonical morphism of G onto G/H. Then the canonical mappings

are Lie group isomorphisms.

By Proposition 30, AH is a Lie subgroup of G. By Corollary 2 to Proposition 29, $H \cap A$ is a Lie subgroup of G. It is therefore meaningful to speak of the groups $AH/H$ and $A/(H \cap A)$. On the other hand, $\phi(A)$, which is the canonical image of AH in G/H, is closed and hence is a Lie subgroup of G/H (Proposition 28 (iii)). Proposition 28, applied to the composite morphisms $A \to G \to G/H$ and $AH \to G \to G/H$, proves that the canonical mappings of the proposition are Lie group isomorphisms.