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I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this? Thanks....

Let $U$ be an $n \times n$ orthogonal matrix. Given an arbitrary partition ${\mathcal P}_c=\{y_1,y_2,\ldots,y_k\}$ of the columns of $U$, does there always exist a corresponding partition ${\mathcal ...