$[s_1, \, s_2]$ are the Dynkin labels of the Lorentz algebra. Combined with the bounds given by the field content, this implies that $D_0$ also satisfies $$\begin{aligned}
 D_0 &=& L-B+s_1 = L+B+s_2.\end{aligned}$$

The ${{\mathfrak{psu}}}(1,1|2)$ algebra \[sec:psu112alg\]
---------------------------------------------------------

We introduce a notation for the subset of the ${{\mathfrak{psu}}}(2,2|4)$ generators that generates the ${{\mathfrak{psu}}}(1,1|2)$ algebra, $$\begin{gathered}
\label{eq:pSU112Gen}
{{\mathfrak{J}}}^0(g)=- {{\mathfrak{L}}}+2{{\mathfrak{D}}}_0+{\delta{\mathfrak{D}}}(g), \quad
{{\mathfrak{R}}}^0={{\mathfrak{R}}}^2_2-{{\mathfrak{R}}}^3_3, \notag \\ {{\mathfrak{J}}}^{++}(g)={{\mathfrak{P}}}_{22}(g),\quad
{{\mathfrak{J}}}^{--}(g)={{\mathfrak{K}}}^{22}(g),\quad
{{\mathfrak{R}}}^{\uparrow\uparrow}(g)={{\mathfrak{R}}}^3_2, \quad
{{\mathfrak{R}}}^{\downarrow\downarrow}(g)={{\mathfrak{R}}}^2_3, \notag \\
\overrightarrow{{{\mathfrak{Q}}}}^{+\downarrow}(g)={{\mathfrak{Q}}}^{2}_2(g),\quad
\overleftarrow{{{\mathfrak{Q}}}}^{-\uparrow}(g)={{\mathfrak{S}}}_{2}^2(g),\quad
\overrightarrow{{{\mathfrak{Q}}}}^{+\uparrow}(g)={{\mathfrak{Q}}}^{3}_2(g),\quad
\overleftarrow{{{\mathfrak{Q}}}}^{-\downarrow}(g)={{\mathfrak{S}}}_{3}^2(g),\notag \\
\overleftarrow{{{\mathfrak{Q}}}}^{+\uparrow}(g)=\dot{{{\mathfrak{Q}}}}_{22}(g),\quad
\overrightarrow{{{\mathfrak{Q}}}}^{-\downarrow}(g)=\dot{{{\mathfrak{S}}}}^{22}(g), \quad
\overleftarrow{{{\mathfrak{Q}}}}^{+\downarrow}(g)=\dot{{{\mathfrak{Q}}}}_{23}(g),\quad
\overrightarrow{{{\mathfrak{Q}}}}^{-\uparrow}(g)=\dot{{{\mathfrak{S}}}}^{32}(g).\end{gathered}$$

${{\mathfrak{L}}}$ is the length operator; it multiplies a state composed of $L$ fundamental fields by $L$. ${{\mathfrak{L}}}$ commutes with all of the ${{\mathfrak{psu}}}(1,1|2)$ generators. In this sector, it satisfies $${{\mathfrak{L}}}={{\mathfrak{L}}}^2_2-\dot{{{\mathfrak{L}}}}^2_2-2{{\mathfrak{R}}}^1_1.$$ ${{\mathfrak{L}}}^\alpha_\beta$ and $\dot{{{\mathfrak{L}}}}^{\dot{\alpha}}_{\dot{\beta}}$ are Lorentz rotations. ${{\mathfrak{D}}}_0$ is the classical dilatation generator, and ${\delta{\mathfrak{D}}}$ is its quantum correction. ${\delta{\mathfrak{D}}}$’s leading term is at $\mathcal{O}(g^2)$, and its expansion includes even powers of $g$ only. The remaining generators appearing on the right side of (\[eq:pSU112Gen\]) are the following ${{\mathfrak{psu}}}(2,2|4)$ generators: ${{\mathfrak{su}}}(4)$ internal R-symmetry rotation generators ${{\mathfrak{R}}}$, Lorentz translations and boosts ${{\mathfrak{P}}}$ and ${{\mathfrak{K}}}$, and fermionic supertranslations and superboosts ${{\mathfrak{Q}}}$, $\dot{{{\mathfrak{Q}}}}$, ${{\mathfrak{S}}}$, and $\dot{{{\mathfrak{S}}}}$. Appendix D of [@Beisert:2004ry] gives a complete description of the full algebra.

Note that the superscript signs correspond to ${{\mathfrak{su}}}(1,1)$ charge 