We improve homological stability ranges for the orthogonal group, special orthogonal group, elementary orthogonal group and the spin group over a commutative local ring $R$ with infinite residue field such that 2 ∈ R*.

The authors thank the anonymous referee for providing very detailed feedback on this paper. The second author also gratefully acknowledges funding from the University of Warwick and the UK Engineering and Physical Sciences Research Council (Grant number: EP/V520226/1) during his time as a PhD student, when this paper was first written.