$G$ is a compact nontrivial Lie group with a bi-invariant metric, is $\{g \in T_eG|4 \geqslant |g|>1\}$ a lie subalgebra?

Question

$G$ is a compact nontrivial Lie group with a bi-invariant metric, is $\{g \in T_eG|4 \geqslant |g|>1\}$ a lie subalgebra?

I doubt that this would be true since it might not be closed under the lie bracket.

Answer 1

Forget the bracket - it is not even a vector subspace! More specifically, if $g\in T_e G$ has length between $1$ and $4$, then $5g$ doesn't.