Classification of the Absolute-Valued Algebras with Left-Unit Satisfying x2(x2)2 = (x2)2x2

TL;DR

This paper classifies finite-dimensional absolute-valued algebras with a left-unit satisfying a specific algebraic identity, identifying known algebras and introducing two new ones.

Contribution

It provides a complete classification of such algebras up to dimension four, including the discovery of two previously unlisted algebras.

Findings

Finite-dimensionality of degree at most 4 for these algebras

Complete list of known algebras satisfying the conditions

Introduction of two new absolute-valued algebras

Abstract

We show that every absolute-valued algebra with left-unit satisfying (x2; x2; x2) = 0 is finite-dimensional of degree at most 4: Next, we determine such an algebras. In addition to the already known algebras R; C; \astC; H; \astH; \astH(i; 1); O; \astO; \astO(i; 1) the list is completed by two new algebras not yet specified in the literature.