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We establish an isomorphism between the Grothendieck-Teichm\"uller Liealgebra $\mathfrak{grt}_1$ in depth two modulo higher depth and the cohomologyof the two-loop part of the graph complex of internally connected graphs$\mathsf{ICG}(1)$. In particular, we recover all linear relations satisfied bythe brackets of the conjectural generators $\sigma_{2k+1}$ modulo depth threeby considering relations among two-loop graphs.  The Grothendieck-Teichm\"uller Lie algebra is related to the zerothcohomology of M. Kontsevich's graph complex $\mathsf{GC}_2$ via T. Willwacher'sisomorphism. We define a descending filtration on $H^0(\mathsf{GC}_2)$ and showthat the degree two components of the corresponding associated graded vectorspaces are isomorphic under T. Willwacher's map.

Filtrations on graph complexes and the Grothendieck-Teichm\"uller Lie algebra in depth two