I am trying to show that if we have $X,Y \in \mathcal{X}(M)$ complete vector fields then for all $s,t$ we have that $\phi_X^t\circ \phi_Y^s=\phi_Y^s\circ \phi_X^t $ if and only if $[X,Y]=0$.
Now trying to show this I am using a result that says that $[X,Y](p)=\frac{d}{d\epsilon}\phi_Y^{-\sqrt \epsilon} \circ \phi_X^{-\sqrt \epsilon} \circ \phi_Y^{\sqrt \epsilon}\circ \phi_X^{\sqrt \epsilon}(p)|_{\epsilon=0}$.
Now assuming the flows commute it's easy to see that $[X,Y]=0$. Now if we assume that $[X,Y]=0$ I can show that for any $t$ we have $\phi_X^t\circ \phi_Y^t=\phi_Y^t\circ \phi_X^t$. Now the idea for proving the case when $s\neq t$ would be to use property of the flows that say that $\phi_X^{t+s}=\phi_X^t\circ \phi_X^s$. Now I have tried to do this but I never got what I wanted. Does anyone know if this approach is going to work ? Thanks in advance.