Associated graded Lie algebra of a filtered group ($\mathrm{gr}G$)

For $G_{\lambda }=\{g:\omega (g)\ge \lambda \}$ and $G_{\lambda +}=\{g:\omega (g)>\lambda \}$, the associated graded groupis a graded Lie algebra with bracket induced by the group commutator. For a p-valuation, $t{\mathrm{gr}}_{\lambda }(g)={\mathrm{gr}}_{\lambda +1}(g^p)$ makes it a graded ${\mathbb{F}}_p[t]$-Lie algebra.