# Commutativity of two vector fields. Prooving that $\frac{d}{dt}\mid_{t=0}\alpha (t)=[X,Y]_{p}$

If $X$ and $Y$ are smooth vector fields with flows $\phi^{X}$ and $\phi^{Y}$, then starting at some $p\in M$, if we flow with $X$ for time $\sqrt t$, then flow with $Y$ for time $\sqrt t$ and then flow backwards along $X$ and $Y$ for time $\sqrt t$, we arrive at a point $$\alpha(t):=\phi^{X}_{-\sqrt t}\circ \phi^{Y}_{-\sqrt t}\circ \phi^{X}_{\sqrt t} \circ \phi^{Y}_{\sqrt t}.$$ It turns out that $\alpha(t)$ is ussualy not $p$. In Lees'"Manifolds and differential geometry" excercise asks to prove the following statement: $$\frac{d}{dt}\mid_{t=0}\alpha (t)=[X,Y]_{p}$$ I tried to change the variables to get rid of the square root: $$\frac{d}{dt}\mid_{t}\phi^{X}_{-\sqrt t}\circ \phi^{Y}_{-\sqrt t}\circ \phi^{X}_{\sqrt t} \circ \phi^{Y}_{\sqrt t}=\frac{d}{2sds}\mid_{s^{2}}\phi^{X}_{-s}\circ \phi^{Y}_{-s}\circ \phi^{X}_{s} \circ \phi^{Y}_{s}$$ But I made no progress with it and do not know how to attack this problem. I would appreciate any hint.

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