Let \([{\mathcal H},(V_1,V_2)]\) be a bi-isometry, i.e., \(V=(V_1,V_2)\) is a pair of commuting isometries on the complex separable Hilbert space \(\mathcal H\). Let \[ \mathcal H=\mathcal H_u\oplus \mathcal H_c,\quad V_1V_2=(V_1V_2)_u\oplus(V_1V_2))_c \] be the Wold decomposition of the product isometry \([\mathcal H, V_1V_2]\). Since the summands \(\mathcal H_u\) and \(\mathcal H_c\) reduce the operators \(V_1\) and \(V_2\) and \(\mathcal H_u\) is the maximal subspace reducing both \(V_1\) and \(V_2\) to unitary operators, we can assume that \(\mathcal H_u=\{0\}\), which means that \(V_1V_2\) is completely non unitary (c.n.u.). Such a bi-isometry is called a c.n.u. bi-isometry. Let \([\mathcal H, V]\) be a c.n.u. bi-isometry. Let \(\mathcal E=\ker(V_1V_2)^*\). It is known that there exists a unitary operator \(W\) and a selfadjoint projection \(P\) on \(\mathcal E\) such that \([\mathcal H, V]\) is unitarily equivalent by Fourier transform to \([H^2(\mathcal E),(V_1(\cdot),V_2(\cdot))]\), where \(V_1(\cdot)\) and \(V_2(\cdot)\) are the “multiplication” operators on \(H^2(\mathcal E)\) with the operator-valued inner functions \(\{\mathcal E, \mathcal E, V_j(\cdot)\}\) defined by \[ V_1(z)=W(zP+P^\perp), \quad V_2(z)=(zP^\perp+P)W^*, \quad z\in{\mathbb T}. \] The triple \(\{\mathcal E, W, P\}\) is called the unitary model of the c.n.u. bi-isometry \([\mathcal H, V]\). This model appeared for the first time in [C. A. Berger, L. A. Coburn and A Lebow, J. Funct. Anal. 27, 51–99 (1978; Zbl 0383.46010)]. In the paper under review, the authors use this model to characterize summands in the Wold decompositions of a bi-isometry and to obtain new proofs of known results concerning such decompositions.