Complexified compactified Minkowski space (${\mathbb{M}}_{\mathbb{C}}=\mathrm{Gr}(2,{\mathbb{C}}^4)$)

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.