  For a field $F$ and an integer $d\geq 1$, 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
$\partial: A \to A$. We obtain several results concerning the kernel of
$\partial$; for instance this kernel is a subalgebra of $A$ that is free of
rank $d$.
