= Vanishing Lie bracket is equivalent to commuting local flows
For smooth <vector fields> $X,Y$, the <local flows> commute wherever both compositions are defined exactly when $[X,Y]=0$. The derivative of the pullback of $Y$ by the $X$ flow is the pullback of $[X,Y]$. A zero bracket therefore makes $Y$ invariant under that flow; uniqueness of solutions to its ordinary differential equation gives commutation. Conversely commutation implies invariance, whose derivative gives the zero bracket. Neither field need have a complete flow.
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