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The present article is motivated by the theorem of Cartan-Dieudonn\'e which states that every orthogonal transformation is a product of reflections. Its purpose is to determine, for each orthogonal transformation, the minimal number of factors in a decomposition into a product of reflections, and to propose an effective algorithm giving such a decomposition. With the orthogonal transformations $g$ of a quadratic space $(V,q)$, it associates couples $(S,\phi)$ where $S$ is a subspace of $V$, anddoi:10.5539/jmr.v9n2p18 fatcat:marj2u6sqbfc3frgb36hx6xsca