A step in proving Lie derivatives of vector fields are Lie brackets

Question

Let $X$ be a vector field on a manifold $M$ and $F^t$ its associated flow at time $t$. We define the Lie derivative of $Y$ along $X$ as $$\mathcal{L}_XY:=\lim_{t\to 0}\frac{DF^{-t}(Y)-Y}{t}.$$ If we want to put in points, we have at $p\in M$, $$\mathcal{L}_XY|_p:=\lim_{t\to 0}\frac{DF^{-t}(Y|_{F^t(p)})-Y|_p}{t}.$$ I want to show the identity $\mathcal{L}_X Y=[X,Y]$, and in some point along the proof, it seems that I need to show the identity $$(*)\lim_{t\to 0}DF^t(\mathcal{L}_XY)=\mathcal{L}_XY$$ in order to show the identity $$\mathcal{L}_X Y=\lim_{t\to 0}\frac{Y-DF^t(Y)}{t}.$$ Is $(*)$ correct, and if so, how do I show it?