Drazin and group invertibility in algebras spanned by two idempotents

For two given idempotents p and q from an associative algebra A, in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfies (pq)ᵐ⁻¹=(pq)ᵐ but (pq)ᵐ⁻²p≠ (pq)ᵐ⁻¹p for some m(≥2). Subsequently, we categorized all the group invertible elements and established an upper bound for Drazin index of any elements in these algebras spanned by $p,q$. Moreover, we formulate a new representation for the Drazin inverse of (α p+q) under two different assumptions, (pq)ᵐ⁻¹=(pq)ᵐ and λ(pq)ᵐ⁻¹=(pq)ᵐ, here α is a non-zero and λ is a non-unit real or complex number.