  We describe an approximation to the in-medium similarity renormalization
group (IMSRG) method in which we include the effects of intermediate three-body
operators arising within nested commutators. As an initial step, we present the
relevant equations for two nested commutators, all of which can be factorized
so that the method scales like the standard IMSRG(2) approximation, enabling
large-scale calculations. We test the accuracy of this approximation scheme,
and apply it to the isotopic chains of carbon, sulfur and nickel isotopic
chains. We obtain an improved description of spectroscopy, and a reduced
dependence on the choice of the valence space. In addition, we provide an
explanation of the relative importance of the diagram topologies included, with
an eye toward assessing the impact of remaining omitted terms.
