(descended from the ${{\mathfrak{su}}}(2,2)$ Lorentz subalgebra). A generator adds dimension equal to ${{{\textstyle\frac{1}{2}}}}$ (-${{{\textstyle\frac{1}{2}}}}$) of the number of its plus (minus) signs. Similarly, vertical arrows correspond to integer ${{\mathfrak{su}}}(2)$ R-charge, and horizontal arrows correspond to half-integer hypercharge, ${{\mathfrak{B}}}$. Throughout this paper we will work in a basis such that hermitian conjugation requires switching signs and reversing arrows simultaneously. Generators without any arrows or a sign are hermitian.

The ${{\mathfrak{psu}}}(1,1|2)$ algebra is given in Appendix \[sec:psu112comm\]. However, many commutators[^7] can be inferred directly from the notation because the three types of charges are conserved. These conservation rules immediately imply that many commutators vanish. Also, ${{\mathfrak{J}}}^0$ measures ${{\mathfrak{su}}}(1,1)$ charge, and ${{\mathfrak{R}}}^0$ measures ${{\mathfrak{su}}}(2)$ charge.

The ${{\mathfrak{psu}}}(1|1)^2$ algebra {#sec:psu11^2alg}
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The ${{\mathfrak{psu}}}(1|1)^2$ algebra is generated by $$\label{eq:psu11^2Gen}
\overrightarrow{{{\mathfrak{T}}}}^+(g)=\dot{{{\mathfrak{Q}}}}^{14}(g),\quad 
\overleftarrow{{{\mathfrak{T}}}}^-(g)=\dot{{{\mathfrak{S}}}}_{14}(g), \quad
\overrightarrow{{{\mathfrak{T}}}}^-(g)={{\mathfrak{S}}}^1_1(g),\quad 
\overleftarrow{{{\mathfrak{T}}}}^+(g)={{\mathfrak{Q}}}^1_1(g).$$ As before, the horizontal arrows correspond to hypercharge. However, these generators carry no ${{\mathfrak{su}}}(1,1)$ charge, and the sign corresponds instead to their commutators with the external automorphism ${{\mathfrak{L}}},$ $$\label{eq:length}
{\left[{{\mathfrak{L}}},{{\mathfrak{T}}}^{\pm}(g)\right]}=\pm {{\mathfrak{T}}}^{\pm}(g).$$

The non-zero commutators are $$\begin{aligned}
\label{eq:psU11^2Alg}
{\left\{\overrightarrow{{{\mathfrak{T}}}}^+(g),\overleftarrow{{{\mathfrak{T}}}}^-(g)\right\}}&={{{\textstyle\frac{1}{2}}}}{\delta{\mathfrak{D}}}(g), & {\left\{\overrightarrow{{{\mathfrak{T}}}}^-(g),\overleftarrow{{{\mathfrak{T}}}}^+(g)\right\}}&={{{\textstyle\frac{1}{2}}}}{\delta{\mathfrak{D}}}(g).\end{aligned}$$ Any commutator between non-conjugate ${{\mathfrak{T}}}$’s vanishes, including the squares of the ${{\mathfrak{T}}}$. The product structure of the full symmetry algebra will be used many times in the rest of this work: 