Question: Let $G$ be a Lie group, how can we associate $$T_e(\text{Aut}(T_eG)) = \text{Der}(T_eG)$$
My motivation for this question is this particular part from my representation theory lecture:
We define a map $$\text{AD}_g: G \to G, \quad h \mapsto ghg^{-1}$$ And define the adjoint map as its derivation at $e \in G$: $$\text{Ad}_g := D(\text{AD}_g)_e: T_eG \to T_eG$$ Which can be viewed as a map $$\text{Ad}: G \to \text{Aut}(T_eG),\quad g \mapsto \text{Ad}_g$$ This is a representation of G and thus implies a representation on the Lie algebra: $$\text{ad} := D(\text{Ad})_e : T_eG \to T_e(\text{Aut}(T_eG)) = \text{Der}(T_eG)$$ The reason for my question arises at the last step
This can be seen here in wikipedia, for example.