  This paper develops a new chain model for the commutative graph complex
$\mathsf{GC}_2$ which takes Lie graph homology as an input. Our main technical
result is the identification of a large contractible complex of (certain)
tadpoles and higher genus vertices of the Feynman transform of Lie graph
homology. Using this result we identify the anti-invarints of Lie graph
homology in genus $2$ with relations between bracketings of conjectural
generators of $\mathfrak{grt}_1$ in depth 2 modulo depth 3, unifying two a
priori disparate appearances of the space of modular cusp forms in the study of
graph homology.
