The octonionic phase space Hopf map

The octonionic Hopf map, expressing $S^{15}$ as an $S^7$ bundle over $S^8$, appears in the twistor transform in 10 dimensions, $S^8$ playing the rôle of the celestial sphere. A symplectic lift to twistor space manifests $Spin(2,10)$ symmetry. The 25-dimensional spinor orbit of $Spin(2,10)$ is an $S^7$ bundle over the phase space of a massless particle.