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The four-complex-dimensional Grassmannian of two-planes in twistor space. Its Plücker embedding is the Klein quadric. An affine chart consists of graphs $W_x=\{(ix\pi ,\pi )\}$. A Lorentzian reality structure on twistor space or an Euclidean reality structure on twistor space selects different four-real-dimensional compact spacetimes inside this same complex manifold.

The meaningful twistor-geometric interpretation of “colinear in $PN$” is that the entire projective line through two distinct projective null twistors lies in projective null twistor space. Mere membership of two points in a projective line would impose no condition, because every pair of distinct points in ${\mathbb{CP}}^3$ spans one.

Indeed, individual nullness givesThis vanishes for every $a,b\in \mathbb{C}$ precisely when $h(X,Y)=0$. Thus $W=\mathrm{span}(X,Y)$ is a maximal totally isotropic two-plane, a real point of Minkowski conformal compactification under the Lorentzian reality structure on twistor space. The two corresponding alpha-planes meet at this unique $W$. Hence their compact intersection is a real spacetime point, and their real null geodesics meet there.

There are two qualifications. The point can be at conformal infinity, so a finite affine meeting is not automatic. If $X,Y$ represent the same projective point instead, their alpha-planes coincide rather than having a unique intersection. If “colinear” is read only as spanning a line in ${\mathbb{CP}}^3$, individual nullness alone does not force a real intersection; the additional orthogonality condition above is indispensable.

For a concrete counterexample to the weaker reading, take $X=(0,0,1,0)$ and $Y=(1,0,0,1)$ in $({\omega }^0,{\omega }^1,{\pi }_{0^{\prime }},{\pi }_{1^{\prime }})$ coordinates. Each is a null twistor, but $h(X,Y)=1$. Their two incidence equations give $x=-i\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, which is not Hermitian. Thus their alpha-planes meet at a complex point that is not a real Minkowski point.

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.