= Normal subgroups of a compact group with irreducible adjoint action
Let $G$ be <connected> and <compact> with nonzero <irreducible> real <Adjoint representation>. Its <Lie algebra> has no proper invariant <Lie algebra ideal>. If it is <Abelian>, irreducibility forces <dimension> one, so $G$ is the <circle group>. Otherwise its centre has zero-dimensional <Lie algebra> and is finite. A proper closed <normal subgroup> likewise has zero-dimensional <Lie algebra>, hence is finite; connectedness of $G$ makes <conjugation> on that finite subgroup trivial, so the subgroup is central. A <Lie group homomorphism> onto a nontrivial <connected> target with <surjective> differential consequently has finite central <group kernel>. The qualifications “proper” and “nontrivial” are essential.
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