For Minkowski space, denoted , the solutions to the twistor equation are of the form
where and are two constant Weyl spinors and is a point in Minkowski space. The are the Pauli matrices, with the indexes on the matrices. This twistor space is a four-dimensional complex vector space, whose points are denoted by , and with a hermitian form
which is invariant under the group SU(2,2) which is a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime.
Points in Minkowski space are related to subspaces of twistor space through the incidence relation
This incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space, denoted , which is isomorphic as a complex manifold to .
Given a point it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of a parametrized by .
The geometric relation between projective twistor space and complexified compactified Minkowski space is the same as the relation between lines and two-planes in twistor space; more precisely, twistor space is
- :=\mathbb {C} ^{4}.}
It has associated to it the double fibration of flag manifolds where is the projective twistor space
and is the compactified complexified Minkowski space
and the correspondence space between and is
In the above, stands for projective space, a Grassmannian, and a flag manifold. The double fibration gives rise to two correspondences (see also Penrose transform), and
The compactified complexified Minkowski space is embedded in by the Plücker embedding; the image is the Klein quadric.