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The fundamental identity of quadratic Jordan algebras $Q_{Q_a b} = Q_aQ_bQ_a$is commonly proven as a consequence of MacDonalds theorem or using moreanalytic methods. In this short note we give a self-contained purely algebraicproof using just a few easily proven identities and a Python script thatfollows a simple randomised logic to reduce expressions of Jordan operators.

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