An algebra generated by two sets of mutually orthogonal idempotents

For a field $F$ and an integer $d\ge 1$, we consider the universal associative $F$-algebra $A$ generated by two sets of $d+1$ mutually orthogonal idempotents. We display four bases for the $F$-vector space $A$ that we find attractive. We determine how these bases are related to each other. We describe how the multiplication in $A$ looks with respect to our bases. Using our bases we obtain an infinite nested sequence of 2-sided ideals for $A$. Using our bases we obtain an infinite exact sequence involving a certain $F$-linear map $\mathrm{\partial }:A\to A$. We obtain several results concerning the kernel of $\mathrm{\partial }$; for instance this kernel is a subalgebra of $A$ that is free of rank $d$.