Constructive and stable cartan–dieudonné and applications to binary quadratic forms over number fields

Abstract We construct an explicit Cartan–Dieudonné decomposition for orthogonal group elements of binary quadratic forms over non-archimedean local fields of characteristic zero, expressing each group element as a product of reflections defined by vectors. A key feature of our construction is its stability: we establish quantitative control on how the reflection matrices vary under small perturbations of the underlying vectors. Using this decomposition, we establish an effective result for the equivalence of binary quadratic forms over number fields. Specifically, let K be a number field and S a finite set of non-archimedean places of K . Given two K -equivalent binary quadratic forms integrally equivalent at every prime in S , we provide an explicit search bound for finding a K -equivalence that are integral at all primes in S .