Lines of three-dimensional real projective space, carried by Plucker coordinates onto Klein's quadric, can be embedded as null vectors of the Clifford algebra Cl(3,3) built on the quadric's own quadratic form. In this homogeneous model every classical line-geometric entity — pencils, bundles and fields of lines, reguli, linear line congruences and linear line complexes — appears as the null space of a blade, and the sandwich action of a grade-1 element is exactly a null polarity of P3(R). The versor group therefore doubles-covers the regular projective transformations, with even versors giving collineations and odd versors correlations, and every regular projective transformation factors as a product of at most six null polarities.
This arxiv paper by Daniel Klawitter presents a Clifford algebraic framework for studying line geometries. It addresses the classical theory of line geometries in projective space, including Plucker …