Let $n\in\mathbb N$, let $M(n,\mathbb C)$ denote the real [vector space](/page/Vector%20Space) and associative algebra of complex $n\times n$ matrices, let $G\le GL(n,\mathbb C)$ be a matrix Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g\subset M(n,\mathbb C)$, and let $H\le G$ be a subgroup that is closed in the [subspace topology](/page/Subspace%20Topology) inherited from $G$. Define

\begin{align*} \mathfrak h:=\{X\in\mathfrak g:\exp(tX)\in H\text{ for every }t\in\mathbb R\}. \end{align*}

Then $\mathfrak h$ is a real Lie subalgebra of $\mathfrak g$; that is, $\mathfrak h$ is a real vector subspace of $\mathfrak g$ and $[X,Y]\in\mathfrak h$ whenever $X,Y\in\mathfrak h$.