Linear transvection groups and embedded polar spaces

Most classical groups arising from (anti-) hermitian forms or (pseudo-) qua dratic forms contain so-called isotropic transvections. The isotropic trans vection subgroups of these classical groups, i.e., the subgroups generated by all isotropic transvections with a fixed axis, form a class Sigma of abe lian subgroups which is a class of abstract transvection groups in the sens e of Timmesfeld [24]. In this paper we give a common characterization of al l these classical groups with isotropic transvections as linear groups gene rated by a class Sigma of abstract transvection groups such that the elemen ts of A is an element of C are transvections.