Spheres with more than 7 vector fields: All the fault of Spin(9)

We give an interpretation of the maximal number of linearly independent vector fields on spheres in terms of the Spin(9) representation on R16. This casts an insight on the role of Spin(9) as a subgroup of SO(16) on the existence of vector fields on spheres, parallel to the one played by complex, quaternionic and octonionic structures on R2, R4 and R8, respectively.