  The Johnson kernel is the subgroup of the mapping class group of a closed
oriented surface that is generated by Dehn twists along separating simple
closed curves. The rational abelianization of the Johnson kernel has been
computed by Dimca, Hain and Papadima, and a more explicit form was subsequently
provided by Morita, Sakasai and Suzuki. Based on these results, Nozaki, Sato
and Suzuki used the theory of finite-type invariants of 3-manifolds to prove
that the torsion subgroup of the abelianized Johnson kernel is non-trivial.
  In this paper, we give a purely 2-dimensional proof of the non-triviality of
this torsion subgroup and provide a lower bound for its cardinality. Our main
tool is the action of the mapping class group on the Malcev Lie algebra of the
fundamental group of the surface. Using the same infinitesimal techniques, we
also provide an alternative diagrammatic description of the rational
abelianized Johnson kernel, and we include in the results the case of an
oriented surface with one boundary component.
