= Lie algebra of a closed subgroup
{c}
{title2=$\mathfrak h=\{X\in\mathfrak g:\exp(tX)\in H\ \forall t\in\mathbb R\}$}
For a closed <subgroup> $H$ of a <Lie group>, this set is a real <Lie subalgebra>. Scalar closure is immediate; the <Lie product formula in a Lie group> and closedness of $H$ give addition. It is a closed <vector subspace>, also directly by continuity of the <Exponential map of a Lie group>. Conjugating its <one-parameter subgroups> by elements of $H$ preserves it. Differentiating the <Adjoint representation of a Lie group> along $\exp(sX)$ gives $[X,Y]\in\mathfrak h$ for $X,Y\in\mathfrak h$. After the <closed-subgroup theorem> establishes smoothness, $\mathfrak h$ is exactly the tangent space $T_eH$.
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