More variations on decomposition of transvections

The method of decomposition of unipotents consists in writing elementary matrices as products of factors lying in proper parabolic subgroups, whose images under inner automorphisms also fall into proper parabolic subgroups of certain types. For the general linear group this method was first proposed by Stepanov in 1987 to simplify the proof of Suslin's normality theorem. Soon thereafter Vavilov and Plotkin generalised it to other classical groups and Chevalley groups. Subsequently, many further results of that type have been discovered. In the present paper we describe new versions of decomposition of unipotents, which allow to expand its applicability well beyond the present scope. Here we merely illustrate these ideas for split classical groups, in some simplest cases. Detailed calculations are relegated to subsequent publications. Bibl. – 34 titles.