Normal subgroups vs ideals of the Lie algebra

Proposition
A closed subgroup $H$ of a compact connected Lie group $G$ is normal iff its Lie algebra $\mathfrak{h}$ is an ideal of the Lie algebra $\mathfrak{g}$.$◼$
Proof
(By the way, in the following proof it is proven that both adjoint representation, the one of the Lie group and the one of the Lie algebra, have a correspondence by means of the exponential map, that is, $$exp(ad(w))=1+ad(w)+\frac{1}{2}ad(w)^2+\cdots=Ad_{exp(w)}.$$)

![[subgruponormal-algebraideal.pdf]]

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