  For $A$ a $C^*$-algebra, $E_1, E_2$ two Hilbert bimodules over $A$, and a
fixed isomorphism $\chi : E_1\otimes_AE_2\to E_2\otimes_AE_1$, 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 $C^*$-algebras.
