Necessary and sufficient conditions for the regularity of the Sylow $p$-subgroups of the Chevalley groups over ${\Bbb Z}_p$ and ${\Bbb Z}_{p^2}$

Let $G$ be an elementary Chevalley group of type $A_n$, $B_n$, $C_n$, and $D_n$ over a finite field of characteristic $p$ or the integer residue ring modulo $p^2$. We show that a Sylow $p$-subgroup $P$ of $G$ is regular if and only if the nilpotency length of $P$ is less than $p$. We introduce and study some series of the combinatorial objects related to the root systems and structure constants of simple complex Lie algebras.