Exercise 5, § 5

Let $\Gamma = \Gamma(X)$ be the Magnus group on $X$ and let $(\Gamma_n)$ be its natural filtration (no. 2). Show that the graded Lie algebra $\mathrm{gr}(\Gamma)$ corresponding to this filtration is isomorphic to the Lie subalgebra $\bigoplus_{n \geq 1} A_n(X)$ of $A(X)$. If $X \neq \emptyset$, this Lie algebra is not generated by its elements of degree 1; deduce that $(\Gamma_n)$ is not the lower central series of $\Gamma$.