Let $X$ be a scheme over an algebraically closed field $k$ and let $\operatorname{Aut}(X)$ denote the functor sending a $k$-scheme $T$ to the group $\operatorname{Aut}_T(X \times_k T)$ of automorphisms of $X \times_k T$ over $T$.

My goal is to have a better grasp of why $\operatorname{Lie}(\operatorname{Aut}(X))= H^0(X, \mathcal{T} X)$ and therefore I am trying to work through an example where I know both the group $\operatorname{Aut}(X)$ and $\operatorname{Lie}(\operatorname{Aut}(X))$.

Let $X = \mathbb{P}_k^1$ so that $\operatorname{Aut}(X)= PGL(2,k)$. Now I try to recover the fact that $\operatorname{Aut}(X)= PGL(2,k)$.

The global sections of $X$ are locally of the form $a_0 + a_1 z + a_2 z^2$ where $z=v/u$ is a choice of homogeneous coordinates on $X$.

Is it possible to go from this description of global sections to the group $\operatorname{Aut}(X)?

My first guess would be to use the exponential map but I don't know how to apply it in such a concrete example.