Classification of the finite-dimensional algebras generated by two tightly coupled idempotents

Let P and Q be idempotents in a real or complex associative algebra and consider the list of products P,Q,PQ,QP,PQP,QPQ,PQPQ,QPQP,⋯ The number of factors is called the order of the product. We say that P and Q are tightly coupled if the list contains two products which take the same value and whose orders differ by at most 1. The main result of the paper is the classification of all algebras which are generated by two tightly coupled idempotents. In other words, we provide a list of algebras such that every algebra generated by two tightly coupled idempotents is isomorphic to exactly one algebra of the list. For example, it follows that up to isomorphisms there are exactly 16 copies of such algebras in which the equality PQP=PQPQ holds.