  The problem of computing the index of a coincidence isometry of the hyper
cubic lattice $\mathbb{Z}^{n}$ is considered. The normal form of a rational
orthogonal matrix is analyzed in detail, and explicit formulas for the index of
certain coincidence isometries of $\mathbb{Z}^{n}$ are obtained. These formulas
generalize the known results for $n\le 4$.
