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The Grothendieck-Teichm\"uller Lie algebra is a Lie subalgebra of a Liealgebra of derivations of the free Lie algebra in two generators. We show thatthe lower central series of the latter Lie algebra induces a decreasingfiltration of the Grothendieck-Teichm\"uller Lie algebra and we study thecorresponding graded Lie algebra. Its degree zero part had been previouslycomputed by the second author. We show that the degree one part is a moduleover a symmetric algebra, which are both equipped with compatible decreasingfiltrations, and we exhibit an explicit lower bound for the associated gradedmodule. We derive from there some information on explicit expression of thedepth 3 part of the depth-graded of the Grothendieck-Teichm\"uller Lie algebra.

On a lower central series filtration of the Grothendieck-Teichm\"uller Lie algebra grt_1