Longer nilpotent series for classical unipotent subgroups

In studying nilpotent groups, the lower central series and other variations can be used to construct an associated Z⁺-graded Lie ring, which is a powerful method to inspect a group. Indeed, the process can be generalized substantially by introducing N^d-graded Lie rings. We compute the adjoint refinements of the lower central series of the unipotent subgroups of the classical Chevalley groups over the field z/pZ of rank d. We prove that, for all the classical types, this characteristic filter is a series of length Θ(d²) with nearly all factors having p-bounded order.