Towards the reverse decomposition of unipotents. II. The relative case

Recently Raimund Preusser displayed very short polynomial expressions of elementary generators in classical groups over an arbitrary commutative ring as products of conjugates of an arbitrary matrix and its inverse by absolute elementary matrices. In particular, this provides very short proofs for description of normal subgroups. In [27] I discussed various generalisations of these results to exceptional groups, specifically those of types $\mathrm{E}_6$ and $\mathrm{E}_7$. Here, I produce a further variation of Preusser's wonderful idea. Namely, in the case of $\mathrm{GL}(n,R)$, $n\ge 4$, I obtain similar expressions of elementary transvections as conjugates of $g\in\mathrm{GL}(n,R)$ and $g^{-1}$ by relative elementary matrices $x\in E(n,J)$ and then $x\in E(n,R,J)$, for an ideal $J\unlhd R$. Again, in particular, this allows to give very short proofs for the description of subgroups normalised by $E(n,J)$ or $E(n,R,J)$ – and thus also of subnormal subgroups in $\mathrm{GL}(n,R)$.