OK, so this is a really silly question. But for the life of me I do not understand how the map $\Phi$ is defined in the proof presented in the first response to the following question: Let $G$ be a Lie group. Show that there is a diffeomorphism $TG \cong G \times T_e G$.
All it says is $\Phi(g,X)=(Lg)_∗(X)$; but my understanding is that a Tangent bundle of a manifold $M$ is $TM=\{(x,v)|x\in M, v\in T_x(M)\}$, and I do not see which $x$ and $v$ are targeted in this definition.
Let $L_g$ the left multiplication by $g$. bu definition, it is a diffeomorphism of $G$ (with inverse $L_{g^{-1}}$). The image by its derivative $L'_g$ of the tangent space at the identity $T_eG$ is therefore the tangent space $T_gG.$ The map $F: G\times T_eG \to TG$ given by $F(g, v)=(g, L'_gv)$ is therefore a diffeomorphism, and in fact an isomorphism of bundles over $G$.