= Local product coordinates for a closed subgroup
{title2=$F(U,V)=\exp U\exp V,\quad\mathfrak g=\mathfrak m\oplus\mathfrak h$}
Choose a <vector subspace> $\mathfrak m$ complementary to the <Lie algebra of a closed subgroup> $H$. The differential of $F$ at $(0,0)$ is $(U,V)\mapsto U+V$, so the <inverse function theorem> makes $F$ a local <diffeomorphism>. Sufficiently small nonzero $U\in\mathfrak m$ cannot satisfy $\exp U\in H$: otherwise <limit directions of a closed subgroup> give a nonzero element of $\mathfrak m\cap\mathfrak h$. Consequently $H$ is locally exactly the coordinate slice $U=0$, proving that it is an <embedded submanifold>.
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