 Solution

In the compact twistor correspondence, a point $p$ of complexified compactified Minkowski space is a two-dimensional subspace $W_p\subset \mathbb{T}$. The correspondence fiber consists of flags $(\ell ,W_p)$ with $\ell$ a one-dimensional subspace of $W_p$. Its image in projective twistor space is precisely $\mathbb{P}(W_p)$, soThis is the twistor line of $p$. In the affine chart the twistor incidence relation gives $W_p=\{(i{p}^{A{A}^{\prime }}{\pi }_{A^{\prime }},{\pi }_{A^{\prime }}):\pi \in {\mathbb{C}}^2\}$. The linear map $\pi \mapsto (ip\pi ,\pi )$ is injective, and projectivizing it gives the same Complex projective space ${\mathbb{CP}}^1$. The subspace argument also covers points at conformal infinity, where that graph chart fails.