An abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra

We construct an abelian quotient of the symplectic derivation Lie algebra ${\mathfrak{h}}_{g,1}$ of the free Lie algebra generated by the fundamental representation of $\mathrm{S}\mathrm{p}(2g,\mathbb{Q})$. More specifically, we show that the weight $12$ part of the abelianization of ${\mathfrak{h}}_{g,1}$ is $1$-dimensional for $g\ge 8$. The computation is done with the aid of computers.