On algebras generated by idempotents

Idempotents (e^2=e) in (nonassociative commutative) algebras (e.g., Jordanalgebras) behave sometimeslike involutions in a group. Indeed, in some(very) interesting cases one can associate an involutive automorphism of the algebra to an idempotent.This topic is related to Griess algebras (and the Monster group), Majorana algebras, Axial algebras (and the Fischer groups) and Miyamoto involutions. We explore mostly algebras generated by two idempotents, having in mind groups, and an extension of the results to algebras generated by finitely many idempotents (when possible). Joint work with Louis Rowen.