  We define trace-like operators on a subspace of the space of derivations of
the free Lie algebra generated by the first homology group $H$ of a surface
$\Sigma$. This definition depends on the choice of a Lagrangian of $H$, and we
call these operators the \emph{Lagrangian traces}. We suppose that $\Sigma$ is
the boundary of a handlebody with first homology group $H'$, and we show that
the Lagrangian traces corresponding to the Lagrangian $\operatorname{Ker} (H
\rightarrow H')$ vanish on the image by the Johnson homomorphisms of the
elements of the Johnson filtration that extend to the handlebody.
