Bounded reduction of orthogonal matrices over polynomial rings

We prove that a matrix from the split orthogonal group over a polynomial ring with coefficients in a small-dimensional ring can be reduced to a smaller matrix by a bounded number of elementary orthogonal transformations. The bound is given explicitly. This result is an effective version of the early stabilisation of the orthogonal K1 functor proven by Suslin and Kopeiko in [39]. Since the similar effective results for special linear and symplectic groups are obtained by Vaserstein in [44], the present paper closes the problem for split classical groups.