Exercise 1, § 2

Let $g$ be a free Lie algebra with basic family $(x_1, \ldots, x_n, \ldots)$. Let $g_n$ denote the subalgebra of $g$ generated by $x_1, \ldots, x_n$ and let $b_n$ denote the smallest ideal of $g_n$ containing $x_n$.

(a) Show that $b_n$ is the submodule of $g_n$ generated by the

where $k \geqslant 0$ and $i_h \leqslant n$ for all $h$.

(b) Show that $g_n = g_{n-1} \oplus b_n$; deduce that $g$ is the direct sum of the $b_n$ for $n \geqslant 1$.

(c) Show, using (a) and (b), that the K-module $g$ is generated by the elements $[x_{i_1}, [x_{i_2}, \ldots, [x_{i_{k-1}}, x_{i_k}] \ldots]]$, where $k \geqslant 0$ and $i_h \leqslant i_k$ for $h < k$.