  Suppose $\mathcal K$ is $N$-dimensional local field of characteristic $p$,
$\mathcal G =\mathop{Gal}(\mathcal K_{sep}/\mathcal K)$, $\mathcal G_{<p}$ is
the maximal quotient of $\mathcal G$ of period $p$ and nilpotent class $<p$ and
$\mathcal K_{<p}\subset \mathcal K_{sep}$ is such that $\mathop{Gal}(\mathcal
K_{<p}/\mathcal K)=\mathcal G_{<p}$. We use nilpotent Artin-Schreier theory to
identify $\mathcal G_{<p}$ with the group $G(\mathcal L)$ obtained from a
profinite Lie $\mathbb F_p$-algebra $\mathcal L$ via the Campbell-Hausdorff
composition law. The canonical $\mathcal P$-topology on $\mathcal K$ is used to
define a dense Lie subalgebra $\mathcal L^{\mathcal P}$ in $\mathcal L$. The
algebra $\mathcal L^{\mathcal P}$ can be provided with a system of $\mathcal
P$-topological generators and its $\mathcal P$-open subalgebras correspond to
all $N$-dimensional extensions of $\mathcal K$ in $\mathcal K_{<p}$. These
results are applied to higher local fields $K$ of characteristic 0 containing
primitive $p$-th root of unity. If $\Gamma =\mathop{Gal}(K_{alg}/K)$ we
introduce similarly the quotient $\Gamma_{<p}=G(L)$, a dense $\mathbb F_p$-Lie
algebra $L^{\mathcal P}\subset L$, and describe the structure of $L^{\mathcal
P}$ in terms of generators and relations. The general result is illustrated by
explicit presentation of $\Gamma_{<p}$ modulo third commutators.
