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 R^16. 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 R^2, R^4 and R^8, respectively. 
Maurizio Parton, Paolo Piccinni
math.DG math.RA