A Lie group $G$ has a unique linear connection $\nabla$ with respect to which every left-invariant vector field is parallel. Specifically, if $\left\{ E_i\right\}$ is a left-invariant frame, then the Christoffel symbols of $\nabla$ are all $0$ if and only if every left-invariant vector field is parallel.
I'm trying to show that the torsion tensor $\tau(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$ is identically $0$ if and only if $G$ is abelian. I know that in this left-invariant frame, we get $\tau(E_i, E_j) = [E_j, E_i]$, so I need to show $[E_i, E_j] = 0$ if and only if $G$ is abelian. I know if $G$ is connected, then $G$ is abelian iff the Lie bracket identically vanishes, but without $G$ being connected I'm not sure how to proceed.