The elements of the orthogonal group \(\Omega_ n(V)\) as products of commutators of symmetries (Q1815010)

Let \(F\) be a field of characteristic not 2 and let \(V\) be an \(n\)-dimensional nondegenerate quadratic space over \(F\). Every element in the commutator subgroup \(\Omega_n(V)\) of the orthogonal group \(O_n(V)\) is a product of commutators of symmetries. The author shows that, if \(F\) is a local field, every element in \(\Omega_n(V)\) is a product of \([n/2]\) such commutators, except a few elements for which \([n/2]+1\) commutators are required.