Decompositions of congruence subgroups of Chevalley groups.

We formulate and prove relative versions of several decompositions known in the theory of Chevalley groups over commutative rings. These decompositions are used to obtain factorizations in terms of subsystem subgroups of type A ℓ and upper estimates of the width of principal congruence subgroups with respect to Tits–Vaserstein generators. [ABSTRACT FROM AUTHOR]