Is a connected simply connected compact Lie group perfect?

Question

My Question: Given a compact Lie group $G$ which is connected and simply connected, is $G$ a perfect group?

i.e. do we have the derived/commutator subgroup $[G,G]=G$? Do we have a more elementary method to show this?

Answer 1

The Lie algebra of a simply-connected compact Lie group is semisimple, hence perfect. This in turn implies that the group is perfect, too.

More generally, the Lie algebra of a compact Lie group is a direct sum $\mathfrak{s}\oplus \mathfrak{a}$ of a semisimple subalgebra $\mathfrak{s}$ and an abelian Lie algebra $\mathfrak{a}$. On the group level the abelian suablgebra corresponds to copies of the torus $S^1=\Bbb R/\Bbb Z$, which are not simply connected. So if we assume that $G$ is simply connected, we have $\mathfrak{a}=0$.