Generic 2-coverings of finite groups of lie type

In [10] it was shown by R.H. Dye that in a symplectic group G :=Sp_{2l}(2^f) = Iso (V ; (.,.)) defined over a finite field of characteristic 2 every element in G stabilizes a quadratic form of maximal or non-maximal Witt index inducing the bilinear form (.,.). Thus G is the union of the two G-conjugacy classes of subgroups isomorphic to O_{2l}^+(2^f ) and O_{2l}^-(2^f ) embedded naturally. In this paper we classify all finite groups of Lie type (G;F) with this generic 2- covering property (Thm. A). In particular, we will show that there exists also an interesting example in characteristic 3, i.e., in the finite group of Lie type G := F_4(3^f ) every element in G is conjuagte to an element of the subgroup B_4(3^f ) < F_4(3^f ) or of the subgroup 3:^3D_4(3^f ) < F_4(3^f).