Combinatorial Construction of Tangent Vector Fields on Spheres

For every odd $n$, on the sphere $S^n$, $\rho(n)-1$ linear orthonormal tangent vector fields, where $\rho(n)$ is the Hurwitz–Radon number, are explicitly constructed. For each $8\times8$ sign matrix, compositions for infinite-dimensional positive definite quadratic forms are explicitly constructed. The infinite-dimensional real normed algebras thus arising are proved to have certain properties of associativity and divisibility type.