\section{Some Harder Facts about Power Series and Expansions}
\label{sec:harder}
MORE: This section needs a detailed look.
The lower central series $G_n$ of $G$ is defined inductively by
setting $G_1\coloneqq G$ and $G_{n+1}\coloneqq (G,G_n)=\{(x,y)\colon
x\in G,\, y\in G_n\}$. It is clear that $G=G_1\rhd G_2\rhd
G_3\rhd \ldots$, and that the quotients $G_n/G_{n+1}$ are Abelian
groups. It is well known that the group commutator $(x,y)$ induces
a structure of a graded Lie ring on $\calL G\coloneqq\bigoplus_n
G_n/G_{n+1}$ (see e.g.~\cite{MagnusKarrassSolitar:CGT}).
Proposition~\ref{prop:StrongerCommutators} implies that the map
$x\mapsto\bar{x}$ maps $G_n$ to $I^n$ and induces a Lie morphism $\calL
G\to\calA(G)$ and hence an algebra morphism $\calU(\bbQ\otimes_\bbZ\calL
G)\to\calA(G)$, where $\calU$ denotes the universal enveloping algebra.
Quillen~\cite{Quillen:OnGrOfAGroupRing} proves that that morphism is in
fact an isomorphism: $\calU(\bbQ\otimes_\bbZ\calL G)\cong\calA(G)$.
Note that if $F=F(x_i)$ is a free group on some set of generators $(x_i)$
and $F_n$ denotes the lower central series of $F$, then $F_1/F_2$ is the
free Abelian group with generators $x_i$ and $F_2/F_3$ is the free Abelian
group with generators $(x_i,x_j)$ for $i