  In this paper we first demonstrate explicitly that the new models of
integrable nearest-neighbour Hamiltonians recently introduced in PRL 125 (2020)
031604 satisfy the so-called free fermion condition. This both implies that all
these models are amenable to reformulations as free fermion theories, and
establishes the universality of this condition. We explicitly recast the
transfer matrix in free fermion form for arbitrary number of sites in the
6-vertex sector, and on two sites in the 8-vertex sector, using a Bogoliubov
transformation. We then put this observation to use in lower-dimensional
instances of AdS/CFT integrable R-matrices, specifically pure Ramond-Ramond
massless and massive AdS_3, mixed-flux relativistic AdS_3 and massless AdS_2.
We also attack the class of models akin to AdS_5 with our free fermion
machinery. In all cases we use the free fermion realisation to greatly simplify
and reinterpret a wealth of known results, and to provide a very suggestive
reformulation of the spectral problem in all these situations.
