The fifteen-parameter conformal group

The space of lines in a Hermitean quadric of signature (2, 2) in complex projective three-space is a quadric of signature (2, 4) in real projective five-space, the conformal compactification of Minkowski space. This geometric fact leads to the classical isomorphism ofPSU(2, 2) and the identity component ofPO(2, 4; ), the 15-parameter conformal group. In this paper it is shown how the geometry and the isomorphism, for all components ofPO(2, 4; ), arise naturally from a real form of the Clifford algebra, and its associated spin groups, of a certain complex vector space determined by skew-symmetric 4×4 matrices and having their Pfaffian as quadratic form.