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Let $(S_1, S_2)$ be a bi-isometry, that is, a pair of commuting isometries$S_1$ and $S_2$ on a complex Hilbert space $\mathscr H.$ By the vonNeumann-Wold decomposition, the hyper-range $\mathscrH_\infty(S_1):=\cap_{n=0}^\infty S^n_1\mathscr H$ of $S_1$ reduces $S_1$ to aunitary operator. Although $\mathscr H_\infty(S_1)$ is an invariant subspacefor $S_2,$ in general, $\mathscr H_\infty(S_1)$ is not a reducing subspace for$S_2.$ We show that $\mathscr H_\infty(S_1)$ reduces $S_2$ to an isometry ifand only if the subspaces $S_2(\ker S^*_1)$ and $\mathscr H_\infty(S_1)$ of$\mathscr H$ are orthogonal. Further, we describe all bi-isometries $(S_1,S_2)$ satisfying the orthogonality condition mentioned above.

Bi-isometries reducing the hyper-ranges of the coordinates