Suppose $H$ is a Lie subgroup of $G$, with corresponding Lie algebras $\mathfrak{h}$ and $\mathfrak{g}$. Let $X \in \mathfrak{g}$ and $Y \in \mathfrak{h}$.
Can we say the following, if so, why? $$ Ad_{e^{tX}}(Y) \in \mathfrak h\implies \left. \dfrac{d}{dt}\right|_{t=0}Ad_{e^{tX}}(Y) \in \mathfrak h.$$