Generators and relations for the unitary group of a skew hermitian form over a local ring

When S is commutative but ⁎ is arbitrary with fixed ring R, an elementary proof that the special unitary group SU(2m,S) is generated by unitary transvections is given. This is used to prove that the reduction homomorphisms SU(2m,S)→SU(2m,S˜) and U(2m,S)→U(2m,S˜) are surjective for any factor ring S˜ of S. The corresponding results for the symplectic group Sp(2m,R) are obtained as corollaries when ⁎ is the identity.