Let $n\ge 1$, let $G\le GL(n,\mathbb C)$ be a matrix Lie group with identity element $e=I_n$ and [Lie algebra](/page/Lie%20Algebra) $\mathfrak g\subset M(n,\mathbb C)$, where $\exp$ denotes the usual matrix exponential, and let $H\le G$ be a subgroup that is closed in the [subspace topology](/page/Subspace%20Topology) inherited from $G$. Then $H$ admits a smooth manifold structure, compatible with its subspace topology inherited from $G$, for which $H$ is an embedded Lie subgroup of $G$ and the inclusion map $H\hookrightarrow G$ is a smooth embedding. Its Lie algebra is $\operatorname{Lie}(H)=\{X\in\mathfrak g:\exp(tX)\in H\text{ for all }t\in\mathbb R\}$.