Exercise 6, § 6

Let $G$ be a torsion-free nilpotent group of finite type. Now prove that, if $n$ is sufficiently large, $G$ can be embedded in the strict lower triangular group of order $n$ over $\mathbf{Z}$. (Let $(i, \bar{G})$ be a $\mathbf{P}$-envelope of $G$, cf. § 5, Exer. 6; endowed with its canonical Lie algebra structure (Exer. 4), $\bar{G}$ is a nilpotent $\mathbf{Q}$-Lie algebra of finite dimension. Apply Ado's theorem (Chap. I, § 7) to $\bar{G}$ and deduce an embedding of $\bar{G}$ in a strict triangular group over $\mathbf{Q}$. Pass from $\mathbf{Q}$ to $\mathbf{Z}$ by conjugating by a suitable integral diagonal matrix.)