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

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