Exercise 6, § 6

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