two-site generator that commutes with ${{\mathfrak{B}}}$, ${{\mathfrak{D}}}_0$ and the ${{\mathfrak{R}}}$’s. Commuting all the generators of the ${{\mathfrak{su}}}(1,1|2) \times {{\mathfrak{psu}}}(1|1)^2$ algebras with a generator such as ${{\mathfrak{y}}}$ maps one solution of the commutation relations to another. It corresponds to the first term in the expansion of the similarity transformation $${{\mathfrak{J}}}\mapsto U {{\mathfrak{J}}}U^{-1}, \quad U = 1 + g^2 {{\mathfrak{y}}}+ \cdots, \quad \mbox{i.e.} \quad {{\mathfrak{J}}}_2 \mapsto {{\mathfrak{J}}}_2 + {\left[{{\mathfrak{J}}}_0,{{\mathfrak{y}}}\right]}.$$ We require ${{\mathfrak{y}}}$ to commute with the ${{\mathfrak{R}}}$’s and with ${{\mathfrak{B}}}$ and ${{\mathfrak{D}}}_0$ to preserve ${\delta{\mathfrak{D}}}$’s manifest R-symmetry and ${\delta{\mathfrak{D}}}$’s eigenstates’ hypercharge and classical dimension assignments. To maintain manifest consistency with the Feynman diagram rules, $U$’s expansion must be in even powers of $n$, consisting of $(\frac{n}{2}+1)$-site to $(\frac{n}{2}+1)$-site interactions. For anti-hermitian (or vanishing) ${{\mathfrak{y}}}$, ${{\mathfrak{X}}}^+$and ${{\mathfrak{X}}}^-$ are hermitian conjugates up to $\mathcal{O}(g^2)$.

Freedom for the $\mathcal{O}(g^2)$ solution
-------------------------------------------

There are two possible sources of freedom for the solution at this order: interactions that vanish on cyclic states (gauge transformations) and homogeneous solutions. We now exclude the former and discuss the latter.

The requirement of even parity rules out the possibility of applying gauge transformations to the solution at this order, since generators are sums of two-site to two-site interactions. This also implies 