Drazin and group invertibility in algebras spanned by two idempotents

For two given idempotents $p\text{ and }q$ from an associative algebra $\mathcal{A},$ in this paper, we offer a comprehensive classification of algebras spanned by the idempotents $p\text{ and }q$. This classification is based on the condition that $p\text{ and }q$ are not tightly coupled and satisfies $(pq)^{m-1}=(pq)^{m}$ but $(pq)^{m-2}p\neq (pq)^{m-1}p$ for some $m(\geq2)\in\mathbb{N}.$ 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 $(\alpha p+q)$ under two different assumptions, $(pq)^{m-1}=(pq)^m$ and $\lambda(pq)^{m-1}=(pq)^m,$ here $\alpha$ is a non-zero and $\lambda$ is a non-unit real or complex number.