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It is known that the non-zero part of compact defect operators ofBerger-Coburn-Lebow pairs (BCL pairs in short) of isometries are diagonaloperators of the form \[ \begin{bmatrix} I_1 & & & \\ & D & & \\ & & - I_2 & \\& & & - D \\ \end{bmatrix}, \] where $I_1$ and $I_2$ are the identity operatorsand $D$ is a positive contractive diagonal operator. We discuss the question ofconstructing an irreducible BCL pair from a diagonal operator of the abovetype. The answer to this question is sometimes in the affirmative and sometimesin 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 ofpairs of commuting isometries.

Pairs of projections and commuting isometries