following Page 107, Fulton/Harris First course in Rep theory:
Let G be a Lie group and $\gamma: I \rightarrow G$ be an arc with $\gamma(0)=e$ and $\gamma'(0)=X$
$[X,Y]=ad(X)(Y) = \frac{d}{dt}|_{t=0}(Ad(\gamma(t))(Y))$
Applying the product rule to $Ad(\gamma(t))(Y) = \gamma(t)Y\gamma(t)^{-1}$
This is:
$$=\gamma'(0) \cdot Y \cdot \gamma(0) + \gamma(0) \cdot Y \cdot (-\gamma(0)^{-1} \cdot \gamma'(0) \cdot \gamma(0)^{-1}) = X \cdot Y - Y \cdot X$$
I'm confused about the above application of the product rule... The first summand makes sense, but then in the second summand, to the right of the $Y$, why is there more than just $\gamma'(0)$? We have $-\gamma(0)^{-1}$ and $\gamma(0)^{-1}$.
Thanks!