The statement-title above is a result that should follow from the hint I got to solve it: "Show that the inversion map $i : G \rightarrow G ~,~ g \longmapsto g^{-1} $ is a Lie group homomorphism if G is commutative, and then study the resulting Lie algebra homomorphism." The first part of this hint was easy, but I am not getting any further. I think the corresponding homomorphism should be the total derivative of $i$ evaluated in the identity of $G$ , but I'm not getting anywhere with this idear yet.
I think the trivial Lie bracket should be $[X,Y] = 0$ for two vector fields on G, which holds iff the corresponding flows of X and Y commute.
Sorry if this post turns out to be strongly related to another - I did not find it. Also sorry if my lack of knowledge about this subject is too big; I'm hoping you can give me some more direction.
Hint: The derivative of $i$ at $e$ is $-id_{\mathfrak g}$; just check what it means that $-id$ is a Lie algebra homomorphism.