Closed subgroup of a semisimple Lie group

Question

If $H\subset{G}$ is a topologically closed subgroup of a compact, connected and semisimple Lie group G, then is $H$ also semisimple? If yes, I need some references where this is stated.

Answer 1

No. In fact every compact semisimple Lie group has at least one non-semisimple closed subgroup: a maximal torus. For a specific example, take the diagonal subgroup of $\mathrm{SU}(2)$.

(A torus is not semisimple since it is abelian, and hence it's Lie algebra has non-trivial solvable ideals.)