  The Grothendieck-Teichm\"uller Lie algebra is a Lie subalgebra of a Lie
algebra of derivations of the free Lie algebra in two generators. We show that
the lower central series of the latter Lie algebra induces a decreasing
filtration of the Grothendieck-Teichm\"uller Lie algebra and we study the
corresponding graded Lie algebra. Its degree zero part had been previously
computed by the second author. We show that the degree one part is a module
over a symmetric algebra, which are both equipped with compatible decreasing
filtrations, and we exhibit an explicit lower bound for the associated graded
module. We derive from there some information on explicit expression of the
depth 3 part of the depth-graded of the Grothendieck-Teichm\"uller Lie algebra.
