  We prove that an element from the Chevalley group of type $E_6$ or $E_7$ over
a polynomial ring with coefficients in a small-dimensional ring can be reduced
to an element of certain proper subsystem subgroup by a bounded number of
elementary root elements. The bound is given explicitly. This result is an
effective version of the early stabilisation of the corresponding
$K_1$-functor. We also give part of the proof of similar hypothesis for $E_8$.
