Let $G \le GL(n,\mathbb C)$ be a matrix Lie group, and define its matrix [Lie algebra](/page/Lie%20Algebra) by
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\begin{align*}
\mathfrak g:=\{X\in M(n,\mathbb C):\exp(tX)\in G\text{ for every }t\in\mathbb R\}.
\end{align*}
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Let $H \le G$ be a subgroup that is closed in the [subspace topology](/page/Subspace%20Topology) inherited from $G$. Define the infinitesimal subgroup of $H$ by
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\begin{align*}
\mathfrak h:=\{X\in\mathfrak g:\exp(tX)\in H\text{ for every }t\in\mathbb R\}.
\end{align*}
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Let $\mathfrak m\subset\mathfrak g$ be a real vector subspace such that
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\begin{align*}
\mathfrak g=\mathfrak h\oplus\mathfrak m.
\end{align*}
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Then there exists an open neighbourhood $V_{\mathfrak m}\subset\mathfrak m$ of $0$ such that