# Compact normal subgroups of connected groups

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?

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 $$G$$.