Lie algebras over rings (Lie rings) are important in group theory. For instance, to every group $G$ one can associate a Lie ring

$$L(G)=\bigoplus _{i=1}^\infty \gamma _i(G)/\gamma _{i+1}(G),$$

where $\gamma _i(G)$ is the $i$-th term of the lower central series of $G$. The addition is defined by the additive structure of $\gamma _i{G}/\gamma _{i+1}(G)$, and the Lie product is defined on homogeneous elements by $[x\gamma _{i+1}(G),y\gamma _{j+1}(G)]=[x,y]\gamma _{i+j+1}(G)$, and then extended to L(G) by linearity.

There are several other ways of constructing Lie rings associated to groups, and there are numerous applications of these. One of the most notorious ones is the solution of the Restricted Burnside Problem by Zelmanov, see the book M. R. Vaughan-Lee, "The Restricted Burnside Problem". There's other books related to these rings, for example,
Kostrikin, "Around Burnside",
Huppert, Blackburn, "Finite groups II",
Dixon, du Sautoy, Mann, Segal, "Analytic pro-$p$ groups".