Dimensional reduction consistent with octonion constraints

Determine a dimensional reduction of the eight-dimensional octonion linear vector space to a lower-dimensional vector space such that the unique algebraic properties of the octonions constrain the resulting vector space structure used to describe the behavior of physical objects.

Background

The paper proposes using the eight-dimensional octonion linear vector space as a framework for coordinatizing space and its tangent structure, highlighting the physical relevance of octonionic algebra. Within this approach, the author notes that accepting the combinatorial roles of scalars, vectors, bivectors, and trivectors as governed by octonion multiplication still leaves unresolved how to reduce the dimensionality while respecting the unique octonionic properties.

The author suggests that the octonion structure should constrain any lower-dimensional vector space used to model physical behavior, but does not specify the reduction procedure or resulting structure. Resolving this requires identifying a mathematically precise reduction consistent with octonion multiplication and its implications for modeling physical objects.