Let $h(t)$ be a smooth curve in a matrix Lie group G, with $h(0)=e$. Let $v \in T_e G$. Then it holds that
$$\left.\frac{d}{dt}\left(Ad_{h(t)}(v)\right)\right|_{t=0} = ad_{\left.\frac{d}{dt} h(t) \right|_{t=0}}(v).$$
... I have been trying to figure out if this holds in a general Lie group. Any hints would be greatly appreciated.
On
I'm not sure what of the wikipedia article I am meant to illustrate... I am a physicist, so I'd "translate" your and the WP language to something 9 out of 10 physicists use in the trenches every week...
$$ h(t)= e^{t(X+ t Y+ O(t^2))}, \qquad h(0)=\mathbb {I}, \qquad h'(0)=X\\ \operatorname{ Ad}_{h(t)} (v) = h(t)~ v ~h^{-1} (t), $$ so that, per WP 's basic Lemma, $$ \operatorname{Ad}_{h(t)} (v) = e^{\operatorname{ad}_{t(X+ t Y+ O(t^2))} } (v) = v + t ~ \operatorname{ad}_X (v) + O(t^2)~. $$ hence $$\left.\frac{d}{dt}\left(\operatorname{Ad}_{h(t)}(v)\right)\right|_{t=0} = \operatorname{ad}_{\left. h'(t) \right|_{t=0}}(v),$$ where $\operatorname{ad}_X (v)= [X,v]$, linear in X, as used above.
I know you'd like this further translated in respectable mathematese, but the connection to the WP proofs and the Rossman text where they came from might be evident.
Yes, it holds in general.
Hint: $\operatorname{ad}=\operatorname{Ad}'(0)$ + chain rule.