simplified since the entire solution is hermitian and given by commutators of leading order generators and ${{\mathfrak{x}}}$. In Appendix \[sec:g2proof\] we verify a representative set of commutators. Here we do one example in detail.

The commutator of the conjugate ${{\mathfrak{J}}}$’s $${\left[{{\mathfrak{J}}}^{--},{{\mathfrak{J}}}^{++}\right]}={{\mathfrak{J}}}^0$$ has the $\mathcal{O}(g^2)$ component $${\left[{{\mathfrak{J}}}^{--},{{\mathfrak{J}}}^{++}\right]}_2={\left[{{\mathfrak{J}}}^{--}_2,{{\mathfrak{J}}}^{++}_0\right]}+ {\left[{{\mathfrak{J}}}^{--}_0,{{\mathfrak{J}}}^{++}_2\right]}={{\mathfrak{J}}}^0_2={\delta{\mathfrak{D}}}_2.$$ Expanding the solution for ${{\mathfrak{J}}}^{++}_2$ yields[^8] $$\begin{aligned}
 {{\mathfrak{J}}}^{++}_2&=&{\left[{{\mathfrak{J}}}^{++}_0,{{\mathfrak{x}}}\right]} \nonumber \\
&=&  {\left\{\overleftarrow{{{\mathfrak{T}}}_1}^-,{\left[\overrightarrow{{{\mathfrak{T}}}_1}^+,{{\mathfrak{j}}}^{++}\right]}\right\}}-{\left\{\overleftarrow{{{\mathfrak{T}}}_1}^+,{\left[\overrightarrow{{{\mathfrak{T}}}_1}^-,{{\mathfrak{j}}}^{++}\right]}\right\}},\end{aligned}$$ where we have defined $${{\mathfrak{j}}}^{++}={\left[{{\mathfrak{J}}}_0^{++},{{\mathfrak{h}}}\right]}.$$ Also, after defining $${{\mathfrak{j}}}^{--}=-{\left[{{\mathfrak{J}}}_0^{--},{{\mathfrak{h}}}\right]},$$ direct computation shows that $${\left[{{\mathfrak{J}}}_0^{--},{{\mathfrak{j}}}^{++}\right]}=-{{{\textstyle\frac{1}{2}}}}{{\mathfrak{L}}}.$$ Using these identities we find, $$\begin{aligned}
 \label{eq:J-J+} {\left[{{\mathfrak{J}}}^{--}_0,{{\mathfrak{J}}}^{++}_2\right]}&=&
{\left\{\overleftarrow{{{\mathfrak{T}}}}^-_1,{\left[\overrightarrow{{{\mathfrak{T}}}_1}^+,{\left[{{\mathfrak{J}}}^{--}_0,{{\mathfrak{j}}}^{++}\right]}\right]}\right\}}-{\left\{\overleftarrow{{{\mathfrak{T}}}_1}^+,{\left[\overrightarrow{{{\mathfrak{T}}}_1}^-,{\left[{{\mathfrak{J}}}^{--}_0,{{\mathfrak{j}}}^{++}\right]}\right]}\right\}} \nonumber \\
&=& - {{{\textstyle\frac{1}{2}}}}{\left\{\overleftarrow{{{\mathfrak{T}}}_1}^-,{\left[\overrightarrow{{{\mathfrak{T}}}_1}^+,{{\mathfrak{L}}}\right]}\right\}}+ {{{\textstyle\frac{1}{2}}}}{\left\{\overleftarrow{{{\mathfrak{T}}}_1}^+,{\left[\overrightarrow{{{\mathfrak{T}}}_1}^-,{{\mathfrak{L}}}\right]}\right\}} \nonumber \\
&=& {{{\textstyle\frac{1}{2}}}}{\left\{\overleftarrow{{{\mathfrak{T}}}_1}^-,\overrightarrow{{{\mathfrak{T}}}_1}^+\right\}}+{{{\textstyle\frac{1}{2}}}}{\left\{\overleftarrow{{{\mathfrak{T}}}_1}^+,\overrightarrow{{{\mathfrak{T}}}_1}^-\right\}} \nonumber \\
&=& {{{\textstyle\frac{1}{2}}}}{\delta{\mathfrak{D}}}_2 \end{aligned}$$ To reach the second to last line, we used (\[eq:length\]), and for the last line (\[eq:psU11\^2Alg\]) was needed. By hermiticity, $${\left[{{\mathfrak{J}}}^{--}_2,{{\mathfrak{J}}}^{++}_0\right]}= {\left[{{\mathfrak{J}}}^{--}_0,{{\mathfrak{J}}}^{++}_2\right]}^\dagger.$$ So, finally we have, $$\begin{aligned}
 {\left[{{\mathfrak{J}}}^{--},{{\mathfrak{J}}}^{++}\right]}_2&=& {\left[{{\mathfrak{J}}}^{--}_0,{{\mathfrak{J}}}^{++}_2\right]}+{\left[{{\mathfrak{J}}}^{--}_2,{{\mathfrak{J}}}^{++}_0\right]} \nonumber \\
&=& {\left[{{\mathfrak{J}}}^{--}_0,{{\mathfrak{J}}}^{++}_2\right]}+{\left[{{\mathfrak{J}}}^{--}_0,{{\mathfrak{J}}}^{++}_2\right]}^\dagger \nonumber \\
&=& {{{\textstyle\frac{1}{2}}}}{\delta{\mathfrak{D}}}_2 + {{{\textstyle\frac{1}{2}}}}{\delta{\mathfrak{D}}}_2^\dagger \nonumber \\
&=& {\delta{\mathfrak{D}}}, \end{aligned}$$ as required by the algebra. We used the hermiticity of ${\delta{\mathfrak{D}}}_2$ to reach the last line.

Order $g^3$ {#sec:g3}
===========

With the $\mathcal{O}(g^2)$ solution, we can use the constraints to find the $\mathcal{O}(g^3)$ solution, which consists of corrections to the ${{\mathfrak{psu}}}(1|1)^2$ generators. We now present and discuss this $\mathcal{O}(g^3)$ solution and its proof, the two-loop dilatation generator that follows, 