Proposition 19

Let G be a topological group, H a Lie group and f a homomorphism of the group G into the group H. Suppose that there exist an open neighbourhood of e_G in G, a chart (V, \phi, E) of the manifold H at e_H and a closed vector subspace F of E admitting a topological supplement, such that f(U) \subset V and \phi \circ f)|U is a homeomorphism of U onto \phi(V) \cap F. Then there exists a unique manifold structure on G such that f is an immersion; this structure is the inverse image under f of the manifold structure on H. With this structure G is a Lie group.

As translations of G (resp. H) are homeomorphisms (resp. analytic isomorphisms), f satisfies condition (R) of Differentiable and Analytic Manifolds, R, 5.8.1. The first two assertions of the proposition then follow from Differentiable and Analytic Manifolds, R, loc. cit. Consider the commutative diagram

where m(x, y) = xy^{-1} (resp. n(x, y) = xy^{-1}) for x, y in G (resp. H). Then n \circ (f \times f) is analytic, hence f \circ m is analytic and hence m is analytic since f is an immersion. Therefore G is a Lie group.

The Lie group structure on G is called the inverse image of the Lie group structure on H under f.