Lie Algebra of Automorphism Group of a Scheme

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.