  We study the abelianization of Kontsevich's Lie algebra associated with the
Lie operad and some related problems. Calculating the abelianization is a
long-standing unsolved problem, which is important in at least two different
contexts: constructing cohomology classes in $H^k(\mathrm{Out}(F_r);\mathbb Q)$
and related groups as well as studying the higher order Johnson homomorphism of
surfaces with boundary. The abelianization carries a grading by "rank," with
previous work of Morita and Conant-Kassabov-Vogtmann computing it up to rank
$2$. This paper presents a partial computation of the rank $3$ part of the
abelianization, finding lots of irreducible $\mathrm{SP}$-representations with
multiplicities given by spaces of modular forms. Existing conjectures in the
literature on the twisted homology of $\mathrm{SL}_3(\mathbb Z)$ imply that
this gives a full account of the rank $3$ part of the abelianization in even
degrees.
