  In this paper we classify the four-qubit states that commute with
$U\otimes{U}\otimes{V}\otimes{V}$, where $U$ and $V$ are arbitrary members of
the Pauli group. We characterize the set of separable states for this class, in
terms of a finite number of entanglement witnesses. Equivalently, we
characterize the two-qubit, Bell-diagonal-preserving, completely positive maps
that are separable. These separable completely positive maps correspond to
protocols that can be implemented with stochastic local operations assisted by
classical communication (SLOCC). This allows us to derive a complete set of
SLOCC monotones for Bell-diagonal states, which, in turn, provides the
necessary and sufficient conditions for converting one two-qubit state to
another by SLOCC.
