Two axes in non-commutative algebras with a Frobenius form

TL;DR

This paper investigates the structure of commutative non-associative algebras with a Frobenius form, focusing on the multiplication properties when two axes of a specific type are present.

Contribution

It provides detailed insights into the multiplication structure of such algebras given two axes of half type, advancing understanding of their algebraic properties.

Findings

Characterization of multiplication in the algebra

Conditions for axes of half type

Structural implications of Frobenius form

Abstract

Throughout this paper $A$ is a commutative non-associative algebra over a field $\mathbb{F}$ of characteristic not $2.$ In addition $A$ posses a Frobenius form. We obtain detailed information about the multiplication in $A$ given two axes of type half in $A.$