  In https://arxiv.org/abs/1411.4887, we found some necessary conditions for a
Riemannian manifold to admit a local limiting Carleman weight (LCW), based upon
the Cotton-York tensor in dimension $3$ and the Weyl tensor in dimension $4$.
In this paper, we find further necessary conditions for the existence of local
LCWs that are often sufficient. For a manifold of dimension $3$ or $4$, we
classify the possible Cotton-York, or Weyl tensors, and provide a mechanism to
find out whether the manifold admits local LCW for each type of tensor. In
particular, we show that a product of two surfaces admits a LCW if and only if
at least one of the two surfaces is of revolution. This provides an example of
a manifold satisfying the eigenflag condition of \cite{AFGR} but not admitting
$LCW$.
