  In the recent study of correlation functions for the infinite XXZ spin chain,
a new pair of anti-commuting operators $b(z), c(z)$ was introduced. They act on
the space of quasi-local operators, which are local operators multiplied by the
disorder operator. For the inhomogeneous chain with the spectral parameters
$\xi_{k}$, these operators have simple poles at $z^2=\xi_{k}^2$. The residues
are denoted by $b_{k}, c_{k}$. At $q=i$, we show that the operators $b_{k},
c_{k}$ are cubic monomials in free fermions. In other words, the action of
these operators is very simple in the fermion basis. We give an explicit
construction of these fermions. Then, we show that the existence of the
fermionic basis is a consequence of the Grassmann relation, the equivariance
with respect to the action of the symmetric group and the reduction property,
which are all valid for the operators $b_{k}, c_{k}$ in the case of generic
$q$.
