Quaternionic Projective Space - in Coordinates

In Coordinates

Its direct construction is as a special case of the projective space over a division algebra. The homogeneous coordinates of a point can be written

where the are quaternions, not all zero. Two sets of coordinates represent the same point if they are 'proportional' by a left multiplication by a non-zero quaternion c; that is, we identify all the

.

In the language of group actions, is the orbit space of by the action of, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside one may also regard as the orbit space of by the action of, the group of unit quaternions. The sphere then becomes a principal Sp(1)-bundle over :

There is also a construction of by means of two-dimensional complex subspaces of, meaning that lies inside a complex Grassmannian.