Associated Lie algebras of p-groups of maximal class

I was reading a paper the other day on Lie algebras of maximal class and they keep saying that some results are taken from p-groups theory.

So my question is how do you get the associated Lie algebra of a p-group? I assume that we get Lie algebras over a finite field, but still how do we get the bracket from the group multiplication? Is it just $[x,y]=xy$?

One can associate to a group $G$ a Lie ring $L(G)$ via the lower central series $G_i$, inductively defined by $G_1=G$ and $G_i=[G,G_{i-1}]$. The associated graded abelian group $$ gr(G) = \bigoplus_{n>1} G_n/G_{n+1} $$ has the structure of a graded Lie algebra over the ring $\mathbb{Z}$ of integers, the bracket operation in $gr(G)$ being induced by the commutator operation in $G$, e.g., by $$ [xG_i,yG_j] = [x,y]G_{i+j}. $$

We can specialise this to finite $p$-groups, replacing $G_{i+1}=[G,G_i]$ by $G_{i+1}=[G,G_i]G_i^p$, obtaining a Lie algebra $L(G)$ over $\mathbb{F}_p$ (because every factor group has exponent $p$).

Reference: Finite $p$-groups in representation theory, section $3$, Lazard correspondence.