Split Octonions and the Rolling Ball

Since $$ PC \cong \mathbb{R}\mathrm{P}^2 \times \mathrm{SU}(2) $$ a point in $PC$ describes the state of a spinorial ball rolling on a projective plane.

The orientation of an ordinary ball is described by an element of $\mathrm{SO}(3)$. For a spinorial ball we use the double cover $\mathrm{SU}(2)$.
So, you need to rotate a spinorial ball twice around any axis to bring it back to its original orientation.
For a ball rolling on a sphere, the point of contact is an element of $S^2$. The projective plane $\mathbb{R}\mathrm{P}^2 $ is $S^2$ with antipodal points identified.
So, a ball rolling on a projective plane is just a ball on a sphere where we decree that rolling it halfway around brings the point of contact back to its original position.