Dimensional reduction in the octonion linear vector space

Determine how the unique algebraic properties of the octonions constrain and effect a dimensional reduction of the eight-dimensional octonion linear vector space (LVS) to a lower-dimensional vector space structure that is appropriate for understanding the behavior of physical objects in space and space-time, including a precise characterization of the resulting manifold and its tangent space.

Background

The paper proposes viewing the eight-dimensional LVS of the octonions as a four-dimensional base manifold with an associated four-dimensional tangent space and explores mappings between these components using octonion algebraic properties, including associators and antiassociators. This perspective aims to clarify pre-metric notions such as orthogonality and mirror symmetry and relates certain null-space structures to classical symmetry-of-aspect ideas due to H. J. S. Smith.

Despite suggesting a specific mapping that splits octonion space into base and tangent parts, the authors explicitly acknowledge that the general issue of dimensional reduction—namely, how octonion algebra uniquely constrains the physically relevant vector space structure—remains unresolved. Establishing this reduction mechanism would provide a rigorous foundation for selecting the appropriate lower-dimensional representation for physical modeling within the octonionic framework.