Are the torsion elements dense in every compact Lie group?

Question

Let $ G $ be a compact connected real Lie group. Denote by $ T $ its set of torsion elements. Is $ T $ always dense in $ G $?

Answer 1

Yes. Here's a sketch:

1. If $G$ is a torus, the set of torsion points of $G$ is dense in $G$.

2. If $H$ is the closure in $G$ of an arbitrary one-parameter subgroup, then (since $G$ is compact) $H$ is a torus. This means the set of torsion points of the subgroup $H$ is (already) dense in $H$. (Of course, it may be that a one-parameter subgroup itself contains only one torsion point!)

3. Since $G$ is compact and connected, every element of $G$ lies in a one-parameter subgroup.