  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. Since
the similar effective results for special linear and symplectic groups were
obtained by Vaserstein, the present paper closes the problem for split
classical groups.
