  The symplectic derivation Lie algebras defined by Kontsevich are related to
various geometric objects including moduli spaces of graphs and of Riemann
surfaces, graph homologies, Hamiltonian vector fields, etc. Each of them and
its Chevalley-Eilenberg chain complex have a $\mathbb{Z}_{\geq 0}$-grading
called weight. We consider one of them $\mathfrak{c}_g$, called the
"commutative case", and its positive weight part $\mathfrak{c}_g^{+} \subset
\mathfrak{c}_g$. The symplectic invariant homology of $\mathfrak{c}_g^{+}$ is
closely related to the commutative graph homology, hence there are some
computational results from the viewpoint of graph homology theory. However, the
entire homology group $H_\bullet (\mathfrak{c}_g^{+})$ is not known well. We
determined $H_2 (\mathfrak{c}_g^{+})$ by using classical representation theory
of $\mathrm{Sp}(2g; \mathbb{Q})$ and the decomposition by weight.
