Explicit parallelizations on products of spheres

Kervaire’s proof does not provide an explicit parallelization on products of spheres. The only reference the author knows to provide explicit parallelizations is [Bru92], that considers the cases when one of the spheres is of dimension 1, 3, 5, 7, and uses some specific arguments of these low dimensions. In [Bru92] the general case is left as an open problem. The aim of this paper is to write an explicit orthonormal parallelization for all parallelizable products of spheres, using an explicit isomorphism with a trivial vector bundle obtained following a hint of [Hir88]. A description of some G-structures on Sm × Sn associated to this parallelization is given in [Par00].