  We give a new approach to the infinitesimal structure of Lipschitz maps into
L^1. As a first application, we give an alternative proof of the main theorem
from an earlier paper, that the Heisenberg group does not admit a bi-Lipschitz
embedding in L^1. The proof uses the metric differentiation theorem of Pauls
and the cut metric decomposition to reduce the nonembedding argument to a
classification of monotone subsets of the Heisenberg group.
