Basic reductions for the description of normal subgroups

Classification of subgroups in a Chevalley group $G(\Phi,R)$ over a commutative ring $R$, normalised by the elementary subgroup $E(\Phi,R)$, is well known. However, for exceptional groups one cannot find in the available literature neither the parabolic reduction, nor the level reduction. This is due to the fact that the Abe–Suzuki–Vaserstein proof relied on localisation and reduction modulo Jacobson radical. Recently for the groups of types $\operatorname{E}_6$, $\operatorname{E}_7$ and $\operatorname{F}_4$ the first-named author, M. Gavrilovich and S. Nikolenko proposed an even more straightforward geometric approach to the proof of structure theorems, similar to the one used for classical cases. In the present work we give still simpler proofs of two key auxiliary results of the geometric approach. First, we carry through the parabolic reduction in full generality: for all parabolic subgroups of all Chevalley groups of rank $\ge 2$. At that we succeeded in avoiding any reference to the structure of internal Chevalley modules, or explicit calculations of the centralisers of unipotent elements. Second, we prove the level reduction, also for the most general situation of double levels, which arise for multiply-laced root systems.