Exercise 1, § 4

Let $(G_n)$ be an integral central filtration on G. Show that the Lie algebra $gr(G)$ is generated by $gr_1(G)$ if and only if $G_n = G_{n+1} \cdot C^n G$ for all $n \geqslant 1$. In that case, show that $G_n = G_m \cdot C^n G$ for $m > n$ and deduce that $(G_n) = (C^n G)$ if there exists an integer m such that $G_m = \{ e \}$.