= Lie algebra of a normal Lie subgroup
{c}
{title2=$H\triangleleft G\Longrightarrow[\mathfrak g,\mathfrak h]\subseteq\mathfrak h$}
Conjugation by every element of $G$ preserves a normal <Lie subgroup> $H$. Its differential therefore preserves $\mathfrak h=T_eH$. Differentiate that invariance along $\exp(tX)$ to obtain $\operatorname{ad}(X)\mathfrak h\subseteq\mathfrak h$, proving that $\mathfrak h$ is an <ideal of a Lie algebra>. Closed subgroups have a canonical Lie-subgroup structure; an arbitrary abstract subgroup need not.
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