Why is the Lie algebra of a group special, and why is the adjoint useful? I'm reading this paper for an introduction to Lie theory, and I'm having trouble understanding the point of the adjoint matrix. But I have a few questions leading up to it.
What makes the Lie algebra of a group, $T_{\epsilon}\mathcal{M}$, special compared to other tangent spaces $T_{\mathcal{X}}\mathcal{M}$ of the group? In other words, what can you do in the Lie algebra that you can't do in another tangent space?
I understand the derivation if $Ad_{\mathcal{X}}$ but I don't fully understand why there needs to be a mapping $^{\mathcal{X}}\tau \mapsto ^{\epsilon}\tau$ to begin with.
Thanks for reading!
EDIT: I've made the question more focused.
 A: Suppose $g\in G$. What is the difference between $T_e G$ and $T_g G$? Well, since inversion and left multiplication by $g$ are smooth, we have linear maps $dL_g:T_e G\to T_g G$ and $dL_{g^{-1}}:T_g G\to T_e G$, and since $g*g^{-1}=e$ these maps re invertible. So each tangent space has a canonical isomorphism to the tangent space at the identity.
Now you might object: the point of manifolds is that we're going to look at vector fields, not just vectors at a point. That is, identifying the tangent spaces doesn't (immediately) help us understand the tangent bundle.
Well, if you give me a vector field $V$ on $G$, I can make new vector fields $gV_{gp} = dL_g(V_p)$ for every element $g\in G$. If $G$ were finite, we could take the average of all of these and get a new vector field, which you can check is left-invariant, i.e., multiplying on the left by any $g\in G$ leaves the vector field unchanged. If $G$ is not finite, but is compact, there is a tool called the Haar measure which you can use to integrate over G and obtain the average. It turns out the left-invariant vector fields on $G$ really captures most (all?) of the information that considering all vector fields does.
So, one tends to work in $T_e G$ and use the left actions to move around and study the local structure, and use the averaging technique to capture the global structure.
