  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.
