Relation between a Lie group and Lie algebra representation for $W \otimes V$

Question

We can define a representation of a Lie group and get the induced representation of the Lie algebra. Let $G$ act on $V$ and $W$, $\mathfrak{g}$ be the Lie algebra associated to $G$ and $X \in \mathfrak{g}$. Then the action on $V$ is defined as $\displaystyle X(v)=\left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)$ where $\gamma_t$ is a path in $G$ with $\gamma'_0 = X$. Then: $$ \begin{align*} X(v \otimes w) & = \left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)\otimes \gamma_t(w) \\ & \stackrel{?}{=}\left(\left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)\right)\otimes w + v\otimes \left(\left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)\right) \\ & = X(v)\otimes w + v \otimes x(w)\end{align*}$$ My question is: why does it split?

Answer 1

$$ \begin{align*} X(v \otimes w) & = \left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)\otimes \gamma_t(w) \\ & = \lim_{t \to 0} \,\,\, \frac{\gamma_t(v)\otimes \gamma_t(w)-v \otimes w}{t} \\ & = \lim_{t \to 0} \,\,\, \frac{\gamma_t(v)\otimes ( \gamma_t(w)-w+w)-v \otimes w}{t} \\ & = \lim_{t \to 0} \,\,\, \frac{\gamma_t(v)\otimes (\gamma_t(w)-w)+ \gamma_t(v)\otimes w-v \otimes w}{t} \\ & = \lim_{t \to 0} \,\,\, \frac{\gamma_t(v)\otimes (\gamma_t(w)-w)+ (\gamma_t(v)-v)\otimes w}{t} \\ & = \lim_{t \to 0} \,\,\, \frac{v \otimes (\gamma_t(w)-w)}{t}+ \lim_{t \to 0} \,\,\, \frac{(\gamma_t(v)-v)\otimes w}{t} \\ & =\left(\left.\frac{d}{dt}\right|_{t=0}\gamma_t(v)\right)\otimes w + v\otimes \left(\left.\frac{d}{dt}\right|_{t=0}\gamma_t(w)\right) \\ & = X(v)\otimes w + v \otimes X(w)\end{align*}$$