Non abelian Lie group with abelian Lie algebra

Question

I've been trying to find an example of a non abelian, non connected Lie group such that its Lie algebra is abelian but I can't find one, so can somenone please help me.

Answer 1

Take any Abelian Lie group $G$ and consider $G\times S_3$. You can use any other non-Abelian discrete group instead of $S_3$. Anyway, the Lie algebra of $G\times S_3$ is (isomorphic to) the Lie algebra of $G$, which is Abelian.

Answer 2

If it's not required to be connected, this is easy. Let $H$ be a finite nonabelian group and consider $S^1\times H$. This has an abelian Lie algebra but is not abelian.