  Let V be a finite dimensional vector space over the two element field. We
compute orbits for the linear action of groups generated by transvections with
respect to a certain class of bilinear forms on V. In particular, we compute
orbits that are in bijection with connected components of real double Bruhat
cells in semisimple groups, extending results of M.Gekhtman, B. Shapiro, M.
Shapiro, A.Vainshtein and A. Zelevinsky.
