Axial identities

Axial algebras are algebras generated by semisimple idempotents. Fusion rules are recalled, with an emphasis on Jordan fusion rules, which include Jordan algebras and Matsuo algebras. Axial algebras can be understood better in terms of idempotental identities and axial identities of axial algebras, which are used to reformulate proofs of major theorems of J. Desmet,  I. Gorshkov,    S. Shpectorov, and A. Staroletov about solid subalgebras. This approach produces generic examples, including an example of a primitive nonsingular axial algebra  of Jordan type 1/2 having radical 0, which is neither Jordan nor a homomorphic image of a Matsuo algebra.