Geodesic
Combinatorial Construction of Tangent Vector Fields on Spheres
Matematičeskie zametki, Tome 83 (2008) no. 4, pp. 590-605

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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.
Keywords: linear orthonormal tangent vector field, odd-dimensional sphere, composition of quadratic forms, Clifford algebra, Hurwitz–Radon theorem, Cayley number. 
@article{MZM_2008_83_4_a10,
     author = {A. A. Ohnikyan},
     title = {Combinatorial {Construction} of {Tangent} {Vector} {Fields} on {Spheres}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {590--605},
     publisher = {mathdoc},
     volume = {83},
     number = {4},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2008_83_4_a10/}
}
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A. A. Ohnikyan. Combinatorial Construction of Tangent Vector Fields on Spheres. Matematičeskie zametki, Tome 83 (2008) no. 4, pp. 590-605. http://geodesic.mathdoc.fr/item/MZM_2008_83_4_a10/