Let $G \le GL(n,\mathbb C)$ be a matrix Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g \subset M(n,\mathbb C)$. Let $\mathfrak h \subset \mathfrak g$ be a real vector subspace, and 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 exist open neighbourhoods $U_{\mathfrak h}\subset \mathfrak h$ of $0$, $U_{\mathfrak m}\subset \mathfrak m$ of $0$, and $U_G\subset G$ of $I$ such that the map