Iterating the Pimsner construction

For $A$ a $C^{\ast }$-algebra, $E_1,E_2$ two Hilbert bimodules over $A$, and a fixed isomorphism $\chi :E_1{\otimes }_A{E}_2\to E_2{\otimes }_A{E}_1$, we consider the problem of computing the $K$-theory of the Cuntz-Pimsner algebra ${\mathcal O}{E_2\otimes_A{\mathcal O}{E_1}}$ obtained by extending the scalars and by iterating the Pimsner construction. The motivating examples are a commutative diagram of Douglas and Howe for the Toeplitz operators on the quarter plane, and the Toeplitz extensions associated by Pimsner and Voiculescu to compute the $K$-theory of a crossed product. The applications are for Hilbert bimodules arising from rank two graphs and from commuting endomorphisms of abelian $C^{\ast }$-algebras.