that the algebra is satisfied exactly (not just modulo gauge transformations).

However, at this point, we cannot rule out modification by a homogeneous solution. Under this modification, $${{\mathfrak{J}}}^{++}_2 \mapsto {{\mathfrak{J}}}^{++}_2 + \delta{{\mathfrak{J}}}^{++}_2, \quad \overleftrightarrow{{{\mathfrak{Q}}}_2}^{+\updownarrow} \mapsto \overleftrightarrow{{{\mathfrak{Q}}}_2}^{+\updownarrow} + \delta \overleftrightarrow{{{\mathfrak{Q}}}_2}^{+\updownarrow},$$ and similarly for the hermitian conjugates. In order for the symmetry constraints to remain satisfied, the $\delta {{\mathfrak{J}}}$’s and $\delta {{\mathfrak{Q}}}$’s must not contribute to any commutator of the algebra. For example, $${\left[\delta{{\mathfrak{J}}}^{++}_2,{{\mathfrak{J}}}^{--}_0\right]}+ {\left[{{\mathfrak{J}}}^{++}_0,\delta{{\mathfrak{J}}}^{--}_2\right]} = 0.$$ We have not found any non-trivial homogeneous solutions, or ruled them out. However, from the above discussion regarding ${\delta{\mathfrak{D}}}$, we conclude that once ${\delta{\mathfrak{D}}}_4$ is found, this freedom is fixed. We will find the solution for ${\delta{\mathfrak{D}}}_4$ below. Since the $\mathcal{O}(g^2)$ solution presented in this section is consistent with it, this is the field theory $\mathcal{O}(g^2)$ solution.

Discussion of the proof of the $\mathcal{O}(g^2)$ solution \[sec:g2discussion\]
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To prove the solution we must check that the commutators given in Appendix \[sec:psu112comm\] are satisfied at $\mathcal{O}(g^2)$. The proof is based upon substitution of the solution, elementary algebra, and keeping track of powers of $g$. The commutators up to $\mathcal{O}(g^1)$ and the identities presented in Appendix \[sec:g2properties\] are also needed. In particular, many of the commutators are 