Closed subgroup theorem

Let $G$ be a Lie group and let $H\le G$ be a closed subgroup.

Theorem (Closed Subgroup Theorem).

There is a unique smooth manifold structure on $H$ making it a Lie group such that the inclusion $\iota:H\hookrightarrow G$ is a smooth injective immersion and a homeomorphism onto its image. In particular, $H$ is an embedded Lie subgroup of $G$ .
The Lie algebra of $H$ is the subalgebra

\mathrm{Lie}(H)=\{X\in \mathfrak{g} : \exp(tX)\in H \text{ for all } t\in \mathbb{R}\},

matching the description in the Lie algebra of a subgroup lemma.

The coset space $G/H$ admits a unique smooth manifold structure such that the projection $\pi:G\to G/H$ is a smooth submersion, making $G/H$ into a basic example of a homogeneous space.

Remarks

Context. This theorem is the bridge between “topological subgroup” and “geometric submanifold.” It is also what makes quotients by closed normal subgroups into Lie groups (compare quotient Lie groups).