$\tilde{\psi}^\alpha$ are the spin connection, the dreibein, and the gravitino $1$-form fields, respectively.[^3]

The corresponding curvature $2$-form $\tilde{F}= d\tilde{A} + \tilde{A} \wedge \tilde{A}$ is[^4] $$\label{curv2f}
\tilde{F} = \tilde{F}^A \tilde{T}_A= \frac{1}{2} \tilde{\mathcal{R}}^{AB} \tilde{J}_{AB} + \tilde{R}^A \tilde{P}_A + \tilde{\Psi}^\alpha \tilde{Q}_\alpha \,  ,$$ with $$\begin{split}
\tilde{\mathcal{R}}^{AB} & \equiv d \tilde{\omega}^{AB} + \tilde{\omega}^A_{\phantom{A} C} \tilde{\omega}^{CB} + \frac{1}{\ell^2} \tilde{V}^A \tilde{V}^B+ \frac{1}{2\ell} \bar{\tilde{\psi}} \Gamma^{AB} \tilde{\psi} = \tilde{R}^{AB} + \frac{1}{\ell^2} \tilde{V}^A \tilde{V}^B+ \frac{1}{2\ell} \bar{\tilde{\psi}} \Gamma^{AB} \tilde{\psi}  \,  , \\
\tilde{R}^A  & \equiv d \tilde{V}^A + \tilde{\omega}^A_{\phantom{A}B} \tilde{V}^B - \frac{1}{2} \bar{\tilde{\psi}} \Gamma^A \tilde{\psi}  = \tilde{T}^A  - \frac{1}{2} \bar{\tilde{\psi}} \Gamma^A \tilde{\psi} \, , \\
\tilde{\Psi} & \equiv \tilde{\nabla} \tilde{\psi} = \tilde{D} \tilde{\psi}  + \frac{1}{2 \ell} \tilde{V}^A \Gamma_A \tilde{\psi} \,  ,
\end{split}$$ where $\tilde{\nabla} = d + \left[ \tilde{A}, \cdot \right]$ and where $\tilde{D}=d + \tilde{\omega}$ is the Lorentz covariant derivative.

The (anti)commutation relations for $\mathfrak{osp}(2|1)\otimes \mathfrak{sp}(2)$ read $$\label{osp}
\begin{split}
& \left[ \tilde{J}_{AB}, \tilde{J}_{CD}\right] =\eta _{BC}\tilde{J}_{AD}-\eta _{AC}\tilde{J}_{BD}-\eta
_{BD}\tilde{J}_{AC}+\eta _{AD}\tilde{J}_{BC} \, ,  \\
& \left[ \tilde{J}_{AB},\tilde{P}_{C}\right] =\eta _{BC}\tilde{P}_{A}-\eta _{AC}\tilde{P}_{B} \,  , \quad  \quad
\left[ \tilde{P}_A , \tilde{P}_B \right] = \frac{1}{\ell^2} \tilde{J}_{AB} \,  ,   \\
& \left[ \tilde{J}_{AB},\tilde{Q}_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{AB}\tilde{Q}\right)_{\alpha } \,  , \quad \quad
\left[ \tilde{P}_{A}, \tilde{Q}_{\alpha }\right] =-\frac{1}{2 \ell} \left( \Gamma _{A}\tilde{Q}\right) _{\alpha } \,  , \\
& 