Let $D\mathcal{F}_{\alpha}: \text{T}G\vert_{e} \to \text{T}G\vert_{\alpha}$ be the differential of a local diffeomorphism $\mathcal{F}$ of the Lie group $G$, where $\alpha \in G$ and $g \in G$. Suppose that $X \in \text{T}G\vert_e$ is a vector field and $\pi : \mathfrak{g} \to \mathfrak{gl}(V)$ be a representation, where $V$ is the vector space $G$ acts on and $\mathfrak{g} \cong \text{T}G\vert_e$ is the Lie algebra of $G$. Under what condition(s) the following statement might be true and $\exists$ a homomorphism of differentials $\Phi$ s.t.:
$$ \pi(D\mathcal{F}_{\alpha}[X])\vert_p = \Phi(D\mathcal{F}_{\alpha})\vert_p[\pi(X\vert_p)], $$ where $p \in V$ and $\vert_p$ “means evaluating relevant mathematical object at point p”. In other words; is it valid to transfer the differential and the vector field through representations separately, then composing them in $V$?