  It is shown that the set of orbits of the action of the elementary symplectic
transvection group on all unimodular elements of a symplectic module over a
commutative ring of characteristic not 2 is identical with the set of orbits of
the action of the corresponding elementary transvection group. This result is
used to get improved injective stability estimates for $K_1$ of the symplectic
transvection group over a non-singular affine algebras.
