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 ${\mathbb{Z}}^{+}$-graded Lie ring, which is a powerful method to inspect a group. Indeed, the process can be generalized substantially by introducing ${\mathbb{N}}^d$-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 $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ of rank $d$. We prove that, for all the classical types, this characteristic filter is a series of length $\mathrm{\Theta }(d^2)$ with nearly all factors having $p$-bounded order.