Partially alternative real division algebras with a few imaginary units

The investigation of four-dimensional partially alternative algebras is further developed in this work. The partial alternativity condition, a natural generalization of the alternativity axiom, broadens the class of alternative algebras, enabling the exploration of non-associative structures with weakened algebraic constraints. We focus on four-dimensional partially alternative unital real division algebras (RDAs) equipped with at least three distinct imaginary units and a reflection. Under these constraints, we achieve a complete classification of the isomorphism classes of such algebras and find that there are, in fact, infinitely many distinct classes. This classification generalizes the classical Frobenius and Hurwitz theorems in the four-dimensional setting. Moreover, we explicitly characterize automorphism groups for these algebras showing that it is either SO(3) if A≅H (the real quaternions) or Z2, otherwise. Since partially alternative algebras do not exist in odd dimensions, and the structure of two-dimensional partially alternative algebras is straightforward, we provide a classification of such algebras up to dimension four.