  It is known that the non-zero part of compact defect operators of
Berger-Coburn-Lebow pairs (BCL pairs in short) of isometries are diagonal
operators of the form \[ \begin{bmatrix} I_1 & & & \\ & D & & \\ & & - I_2 & \\
& & & - D \\ \end{bmatrix}, \] where $I_1$ and $I_2$ are the identity operators
and $D$ is a positive contractive diagonal operator. We discuss the question of
constructing an irreducible BCL pair from a diagonal operator of the above
type. The answer to this question is sometimes in the affirmative and sometimes
in the negative. This also answers a part of the question raised by He, Qin,
and Yang. Our explicit constructions of BCL pairs yield concrete examples of
pairs of commuting isometries.
