Differential of the inverse map gives rise to the bracket of vector fields being zero

Question

Suppose that $J:G\to G$ is the inverse map on a Lie group $G$ And suppose that $X,Y$ are vector fields. We know that if $\phi:G\to G$ is an arbitrary diffeomorphism then $d\phi[X,Y]=[d\phi X, d\phi Y]$. Hence since $J$ is a diffeomorphism this relation holds giving rise to the bracket of left invariant vector fields being zero whether $G$ is abelian or not. I know I am wrong but what is wrong with my argument?