Assume $G$ is a compact Lie group of finite dimension with bi-invariant Riemannian metric. Is the diameter of $G$ bounded above by some constant that can be determined? Alternatively, does there exists a bi-invariant metric on $G$ such that its diameter is bounded above by the diameter of some orthogonal group?
Thoughts:
$G$ being compact Lie group means that $G$ can be embedded as a subgroup in $O(k)$ for some $k$.
We can give $O(k)$ a bi-invariant Riemannian metric and that restricts to bi-invariant metric on $G$, provided the metric restricts to a bi-invariant one on G.