Geometry over composition algebras: Projective geometry

The purpose of this article is to introduce projective geometry over composition algebras: the analogue of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in correspondence with Jordan algebras and that the points of a projective space correspond to rank one matrices in the Jordan algebra. A second part thus studies properties of rank one matrices. I also give an explicit description of the simply-connected Chevalley group of type E 6 over the integers.