An algebraic semi-automated proof of the fundamental identity of Jordan algebras

Abstract

The fundamental identity of quadratic Jordan algebras $Q_{Q_{a}b}=Q_a{Q}_b{Q}_a$ is commonly proven as a consequence of MacDonalds theorem or using more analytic methods. In this short note we give a self-contained purely algebraic proof using just a few easily proven identities and a Python script that follows a simple randomised logic to reduce expressions of Jordan operators.

4 pages + 13 page explanation of the script and code listing