Restriction of adjoint map for normal subgroup of Lie group Let $G$ and $H$ be Lie groups, such that $H$ is a normal subgroup of $G$. Let $\mathfrak{g},\mathfrak{h}$ denote the corresponding Lie algebras. This means that
$$ \forall g\in G, h \in H, ghg^{-1} \in H.$$
In particular, since $\forall X\in\mathfrak{g}, e^{tX} \in G$, we have
$$ \forall X \in \mathfrak{g}, h \in H, e^{tX}h e^{-tX} \in H.$$
Does it follow from this, that
$$\forall X\in\mathfrak{g}, Y\in\mathfrak{h}, e^{tX}Ye^{-tX} \in \mathfrak{h}?$$
Phrased differently, can one restrict the adjoint map $\text{Ad}_{e^{tX}} : \mathfrak{g}\rightarrow\mathfrak{g}$ to $\mathfrak{h}$? I feel like the answer is yes, and the explanation is simple, but I am not seeing it, so perhaps it is not true?

Let $g\in G$, it defines a map $Ad_g:H\rightarrow H$ defined by $Ad_g(h)=ghg^{-1}$.
The differential of $Ad_g$ at $e\in H$ ($e$ is the neutral element) is the adjoint map defined $dAd_g:{\cal H}\rightarrow {\cal H}$ where ${\cal H}$ is the Lie algebra of $H$.
