I have the following question: Let $M$ be a smooth manifold and let $p \in M$. Furthermore let $X$ and $Y$ be two vector fields in a neighbourhood $U$ of $p$ and consider their flows $\varphi^{X}(t,x)$ and $\varphi^{Y}(t,x)$. Define now $s(t):= \varphi^{X}_{\sqrt{t}} \circ \varphi^{Y}_{\sqrt{t}} \circ \varphi^{X}_{-\sqrt{t}} \circ \varphi^{Y}_{-\sqrt{t}}$. How can one show that $\dot{s}(0) = [X,Y]|_{0}$ ? Or is this statement true? If yes, why? Thanks in advance.
Eric