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Let $V$ be a vector space endowed with a non-degenerate quadratic form $Q$.If the base field $\mathbb{F}$ is different from $\mathbb{F}_2$, it is knownthat every isometry can be written as a product of reflections. In thisarticle, we detail the structure of the poset of all minimal length reflectionfactorizations of an isometry. If $\mathbb{F}$ is an ordered field, we alsostudy factorizations into positive reflections, i.e., reflections defined byvectors of positive norm. We characterize such factorizations, under thehypothesis that the squares of $\mathbb{F}$ are dense in the positive elements(this includes Archimedean and Euclidean fields). In particular, we show thatan isometry is a product of positive reflections if and only if its spinor normis positive. As a final application, we explicitly describe the poset of allfactorizations of isometries of the hyperbolic space.

Factoring isometries of quadratic spaces into reflections