Linear transvection groups and embedded polar spaces

Most classical groups arising from (anti-) hermitian forms or (pseudo-) quadratic forms contain so-called isotropic transvections. The isotropic transvection subgroups of these classical groups, i.e., the subgroups generated by all isotropic transvections with a fixed axis, form a class D of abelian subgroups which is a class of abstract transvection groups in the sense of Timmesfeld [24]. In this paper we give a common characterization of all these classical groups with isotropic transvections as linear groups generated by a class D of abstract transvection groups such that the elements of A]D are transvections.