Nilpotence criterion for an abelian-by-cyclic group (${\mathbb{Z}}^r{⋊}_B\mathbb{Z}\text{ is nilpotent}⟺B-I\text{ is nilpotent}$)

For $tv{t}^{-1}=Bv$, the commutator $[t,v]$ is $(B-I)v$ in additive notation. The lower central series from its second term is $(B-I){\mathbb{Z}}^r,(B-I{)}^2{\mathbb{Z}}^r,\dots$. It terminates exactly when $B-I$ is a nilpotent matrix. For rank two this means $(B-I{)}^2=0$, equivalently $\det B=1$ and $\mathrm{tr}B=2$. Nilpotence of a finite-index subgroup alone gives virtual nilpotence of the whole group.