If $G$ is a nilpotent group and $[G,G]$ has Hirsch length $1$, then every f.g. submonoid of $G$ is boundedly generated, i.e. a product of cyclic submonoids. Using a reduction of Bodart, this implies the decidability of the submonoid membership problem for nilpotent groups $G$ where $[G,G]$ has Hirsch length $2$.

翻译：若 $G$ 为幂零群且 $[G,G]$ 的 Hirsch 长度为 $1$，则 $G$ 的每个有限生成子幺半群均为有界生成，即循环子幺半群的乘积。利用 Bodart 的约化，这一结论蕴含了当 $[G,G]$ 的 Hirsch 长度为 $2$ 时幂零群 $G$ 的子幺半群成员问题的可判定性。