Lie algebra of a subgroup lemma

The differential at the identity

is injective, and its image identifies $\operatorname{Lie}(H)$ with the subspace $T_eH\subseteq T_eG$. Under this identification, $\operatorname{Lie}(H)$ is a Lie subalgebra of $\operatorname{Lie}(G)$.

Equivalently: the Lie algebra of a Lie subgroup is its tangent space at the identity, with the bracket inherited from $\operatorname{Lie}(G)$.

When $H$ is a closed subgroup, the closed subgroup theorem guarantees that $H$ is an embedded Lie subgroup, so $T_eH$ is literally a subspace of $T_eG$ in the usual embedded-submanifold sense. In that case, this lemma is the key step behind the subgroup–subalgebra bridge used in the Lie correspondence.