Exercise 13, § 1

Let $g$ be a Lie algebra and $a$ and $b$ two submodules of $g$. For every integer $i \geqslant 0$ we define the submodules $m_i = m_i(a, b)$ and $n_i = n_i(a, b)$ as follows: $m_1 = b, m_{i+1} = [m_i, b], n_1 = a, n_{i+1} = [n_i, b]$. Show that

for $i = 1, 2, \ldots$ (argue by induction on $i$, observing that and $n_i([a, b], b) = n_{i+1}(a, b)$).