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We investigate octonion product deformations coming from the parallelizabletorsion of the 7-sphere $S^7$, obtaining a family of geometries from solutionsof the Lagrangian formalism movement equations. This can be achieved byanalyzing the spontaneous compactification $M_4\times S^7$, where $M_4$ is aLorentzian $4$-dimensional manifold. Besides the usual Riemannian geometry andtwo others proposed by Cartan and Schouten, solutions in geometries withtorsion and more general seven-dimensional spaces are obtained. Such formalismmay by subsequently derived over the 7-sphere $S^7$, locally given by thestructure constants of a nonassociative geodesic loop. Furthermore,$G_2$-structures are investigated, giving rise to the octonion product andbundle $\mathbb{O} M$ over a seven-dimensional manifold $M$. Then, sections ofthis bundle over such space can be perceived as spinor fields in an isometricidentification mapping the spin connection to an octonion covariant derivativepreserving the octonion product defined over $\mathbb{O} M$.

Affinely Connected Spaces, Geodesic Loops, $G_2$-Structures and Deformations