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We know that for a connected locally compact group $G$ there exists a compact normal subgroup $K$such that $G/K$ is a Lie group. Now, if $G$ is compact, could we find a proper compact normal subgroup $K$ with that property?

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The answer is no when $G=\{1\}$, and yes otherwise, as a consequence of the Peter-Weyl theorem. The latter ensures that $G$ has a nontrivial finite-dimensional representation $G\to U(k)$. The image of the latter is a nontrivial Lie quotient of $G$.

More precisely, a consequence of the Peter-Weyl theorem is that every compact group is projective limit of its Lie quotients.

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