If a Lie group is abelian, its Lie algebra has Lie bracket = zero

[ModusPonens]

Hello

I'm looking for tips (and just tips) on how to prove the proposition in the title: If a Lie group is abelian, its Lie algebra has Lie bracket = zero. I know I have to prove that the Lie bracket of the left invariant vector fields (to which the Lie algebra is isomorphic) is zero. But that is as far as I can get. I can't find a way to relate the commutativity of the Lie group with the Lie bracket.

Thanks in advance.