Twistor Hermitian form ($h(Z,W)={\omega }^A{\overline{\rho }}_A+{\pi }_{A^{\prime }}{\overline{\eta }}^{A^{\prime }}$)

For $Z=(\omega ,\pi )$ and $W=(\eta ,\rho )$ the displayed Hermitian form has signature $(2,2)$. It is zero on every pair of vectors in $W_x=\{(ix\pi ,\pi )\}$ when $x$ is Hermitian, that is, Lorentzian real. Conversely a transverse totally isotropic two-plane is such a graph with $x$ Hermitian. This gives the Lorentzian reality structure on twistor space.

Real compactified Minkowski points are the two-dimensional totally isotropic subspaces for the signature-$(2,2)$ twistor Hermitian form. In a graph chart this is the usual Hermitian reality condition on $x^{A{A}^{\prime }}$. Their twistor lines lie in projective null twistor space. On the Klein quadric the same real form is the projectivized null cone of a real quadratic form of signature $(2,4)$.

A nonzero twistor null for the twistor Hermitian form. Its projective class determines a real null geodesic in compactified Minkowski space. For nonzero $\pi$, its affine real solutions of the twistor incidence relation have the form $x=x_0+r{\overline{\pi }}^A{\pi }^{A^{\prime }}$, $r\in \mathbb{R}$. Twistors with $\pi =0$ encode generators at infinity rather than finite affine solutions.

The five-real-dimensional space of projective null twistors. A projective line lies entirely in it precisely when its corresponding two-plane is totally isotropic for the twistor Hermitian form. Such lines represent real points of Minkowski conformal compactification. Two independent null twistors lie on one such line exactly when $h(X,Y)=0$.