A Note on the Construction of Complex and Quaternionic Vector Fields on   Spheres
  A relationship between real, complex, and quaternionic vector fields on spheres is given by using a relationship between the corresponding standard inner products. The number of linearly independent complex vector fields on the standard $(4n-1)$-sphere is shown to be twice the number of linearly independent quaternionic vector fields plus $d$, where $ d = 1 \mbox{ or } 3$. 
Mohammad Obiedat
math.KT