The Lie exponential map and commuting elements

The Lie exponential map and commuting elements

By Baker–Campbell–Hausdorff formula, if $[v,w]=0$ for $v,w$ in the Lie
algebra $\mathfrak g$ of a Lie group $G$ then $exp(v)$ and $exp(w)$
commute in $G$.
Does anyone know a reference or a method of proof of the following partial
inverse: For $v,w$ sufficiently close to $0$ in $\mathfrak g$,
$exp(v)exp(w)=exp(w)exp(v)$ implies $[v,w]=0$?
Being sufficiently close to zero is of course necessary here.