Why lower central series imply $N$ series? Let $G$ be a group and let $H_1,H_2,\cdots$ be subgroups of $G$ such that $$G=H_1\supset H_2\supset\cdots,$$ where $H_{k+1}$ is normal in $H_k$ and $[H_k,H_l]\subset H_{k+l}$ for any $k,l\geq 1$. Such a sequence $\{H_k\}$ is called an $N$-series of $G$. 
I read somewhere that the most familiar example of an $N$-series of $G$ is its lower central series $\Gamma_k$ defined by $\Gamma_1=G$ and $\Gamma_{k+1}=[\Gamma_k,G]$. My question is:
Why the lower central series of a group is an $N$-series? For example, for $k=2$, is it trivial that $[[G,G],[G,G]]$ is a subset of $[[[G,G],G],G]$?
 A: This is not too difficult but I believe it's also not completely trivial.
For three subgroups $H, K, L$ write $[H,K,L] = [[H,K],L]$ (similarly for commutators of elements). Recall the three subgroups lemma:

If two of the subgroups $[H,K,L]$, $[K,L,H]$ and $[L,H,K]$ are trivial, then all three of them are.

We can use this to prove $[\Gamma_i, \Gamma_j] \leq \Gamma_{i+j}$ by induction on $i$. 
The case $i = 1$ is just the definition, so assume $i > 1$. Then $[\Gamma_i, \Gamma_j] = [\Gamma_{i-1}, G, \Gamma_j]$. Here $[G, \Gamma_j, \Gamma_{i-1}]$ and $[\Gamma_j, \Gamma_{i-1}, G]$ are contained in $\Gamma_{i+j}$ by induction. Applying the three subgroups lemma to $G/\Gamma_{i+j}$, we see that the same is true for $[\Gamma_{i-1}, G, \Gamma_j] = [\Gamma_i, \Gamma_j]$.
The three subgroups lemma is an easy consequence of the Hall-Witt identity (here $g^h = h^{-1}gh$).
$$[x,y^{-1},z]^y[y,z^{-1},x]^z[z,x^{-1},y]^x = 1$$
You can use it to give the same proof. For example when $i = j = 2$: 
If $y, z$ are arbitrary elements of $G$ and $x$ is a commutator, the identity shows that $[y,z^{-1}, x]$ is in $\Gamma_4$. So it follows that $[[x,y], [z,w]] \in \Gamma_4$ for any $x, y, z, w \in G$. From this it will follow that $[[G,G], [G,G]] \leq \Gamma_4$ (you might want to prove this).
