How the quotients of central series forms a Lie algebra Let $G$ be a group. Then the subgroups of the central series are $\gamma_1(G)=G$, $\gamma_2(G)=[\gamma_1(G),G],..., \gamma_n(G)=[\gamma_{n-1}(G),G]$. Then in the reference paper given below it is given that the direct sum $\bigoplus_{n\geq1}\frac{\gamma_n(G)}{\gamma_{n+1}(G)}$ forms a Lie algebra over $\mathbb{Z}$. (http://emis.impa.br/EMIS/journals/GMJ/vol9/v9n4-9.pdf).
My first question is that if the notion of Lie algebra is defined over some field, then how we can use $\mathbb{Z}$ (set of integers)?
Kindly explain how this direct sum can become a Lie algebra. In the reference paper above it is given that the commutator map $\gamma_n \times \gamma_m\rightarrow \gamma_{m+n}$ is given by $(x,y)\longmapsto [x,y]$ but I can't understand this mapping.
One more thing: it is given that there is a canonical map from $G_{ab}$ to $\bigoplus_{n\geq1}\frac{\gamma_n(G)}{\gamma_{n+1}(G)}$ inducing a surjective homomorphism from $L(G_{ab})$ to $\bigoplus_{n\geq1}\frac{\gamma_n(G)}{\gamma_{n+1}(G)}$, where $L(G_{ab})$ is the free Lie algebra over $G_{ab}$. How we can define this when $\gamma_2$ is present in the first summand of the direct sum?
I know these are very silly questions but I tried a lot to understand this but was not able to understand. Please help me out to know about this as I am a newcomer to the field of Lie algebras. 
 A: Re first question: Although very often Lie algebras are defined over fields, one can just extend the definition to any (e.g.) commutative ring $R$:

A Lie algebra over $R$ is an $R$-module $L$ with a Lie bracket $[\cdot, \cdot]: L \times L \rightarrow L$ which is bilinear, alternating ($[x,x]=0$ for all $x\in L$) and satisfies  the Jacobi identity.

E.g. Bourbaki only restricts to $R$ being a field after paragraph 3 in the first chapter of their Lie Groups and Lie Algebras book. Also, Lie algebras over the $p$-adic integers $\mathbb Z_p$ are quite common in the theory of $p$-adic analytic groups (Lazard etc.). As mentioned in a comment, sometimes this more general object is called a "Lie ring". Of course one has to be careful about what "standard" results still hold true for this more general definition. 
Re second question: First of all, I take it that for elements $x,y \in G$, the (group) commutator $[x,y]$ is defined as $xyx^{-1}y^{-1}$ (sometimes one defines it as $x^{-1}y^{-1}xy$; insignificant changes apply).
Now let's check the above definition.
A. $\bigoplus \gamma_n(G)/\gamma_{n+1}(G)$ is a $\mathbb Z$-module. Clear, because every summand, being an abelian group, is a $\mathbb Z$-module (write the group operation additively as $\bar x +\bar y := \overline{xy}$).
B. Defining the Lie bracket: First, let's define the Lie bracket on a pair of homogeneous elements (i.e. elements which are contained in one of the summands). This is done as you say: if $\bar x \in \gamma_m(G)/\gamma_{m+1}(G), \bar y \in \gamma_n(G)/\gamma_{n+1}(G)$, then we define the Lie bracket $[x,y]$ as the residue class, in $\gamma_{m+n}(G)/\gamma_{m+n+1}(G)$, of the group commutator $[x,y]$. One checks that this is well-defined, $\mathbb Z$-bilinear, alternating, and satisfies the Jacobi identity.
Now, since every element of the direct sum is a finite sum of homogeneous elements, there is only one way to extend the definition bilinearly to all elements, namely
$$[\sum \bar x_i, \sum \bar y_j] = \sum_{i,j} [\bar x_i, \bar y_j]$$
($\bar x_i \in \gamma_i(G)/\gamma_{i+1}(G)$ etc.) and this is easily checked to still be alternating and satisfy the Jacobi identity.
Re third question: The abelianisation $G_{ab}:= G/[G,G]$ literally is $\gamma_1(G)/\gamma_2(G)$, the first summand in the direct sum, so the "canonical map" is indeed just the inclusion of that first summand into the entire sum. That this induces (or: "lifts to") a homomorphism from the free Lie algebra $L(G_{ab})$ to the sum is part of the universal property of a free object (or whatever definition of free Lie algebra you use, it should follow immediately from that). That it is surjective comes from the fact that additively, the direct sum is generated by all iterated group commutators $[g_1,[g_2,[g_3, .... ]]]$, and these are in the image of the free Lie algebra $L(G_{ab})$. Notice that for example there are generators $h$ of $\gamma_{2}(G)/\gamma_3(G)$ for which there are $x,y \in G$ such that $[x,y] + \gamma_3(G) = h$, and if $x_{ab}, y_{ab} \in G_{ab}$ are the respective residue classes in $G_{ab}$, then in $L(G_{ab})$ there is the element $[x_{ab},y_{ab}]_{L(G_{ab})}$ which via the univeral homomorphism gets mapped to $h$.
