Factorizations of idempotent operator as products of two idempotents

For two commutative idempotents Π1 and Π2, Π1+Π2−Π1Π2 is clearly idempotent. By studying the characterisation of two idempotents generated from factorizations of idempotent operator, we give a necessary and sufficient condition for when there is no another idempotent operator Π3 which has the same range with Π1Π2 such that Π1+ Π2− Π3is idempotent