P_{b}\right] = - \delta_{ab} H \, , \\
& \left[ P_a , P_b \right] = \frac{1}{\ell^2} J_{ab} \, , \quad \quad
\left[ P_a , H \right] = \frac{1}{\ell^2} K_{a} \, , \\
& \left[ J_{ab},Q_{\alpha }\right] =-\frac{1}{2}\left( \Gamma _{ab} Q \right)_{\alpha } \, , \quad \quad
\left[ P_{a}, Q_{\alpha }\right] =-\frac{1}{2 \ell} \left( \Gamma _{a}Q\right) _{\alpha } \, , \\
& \left\{ Q_{\alpha }, Q_{\beta }\right\} = - \frac{1}{\ell} \left(\Gamma^{a0} C \right)_{\alpha \beta} K_{a} + \left( \Gamma ^{0}C\right) _{\alpha \beta }H \, .
\end{split}$$ The corresponding connection $1$-form and curvature $2$-form respectively read $$\label{connadscarrollsuper}
A = \frac{1}{2} \omega^{ab} J_{ab} + k^a K_a + V^a P_a + h H + \psi^\alpha Q_\alpha \,  , \quad
F = \frac{1}{2} \mathcal{R}^{ab} J_{ab} + \mathcal{K}^a K_a + R^a P_a + \mathcal{H} H + \Psi^\alpha Q_\alpha \, ,$$ with $$\label{curvadscarrollsuper}
\begin{split}
\mathcal{R}^{ab} & = d \omega^{ab} + \frac{1}{\ell^2} V^a V^b = R^{ab} + \frac{1}{\ell^2} V^a V^b  \, , \\
\mathcal{K}^a  & = d k^a + \omega^a_{\phantom{a} b} k^b + \frac{1}{\ell^2} V^a h + \frac{1}{2 \ell} \bar{\psi} \Gamma^{a0} \psi = \mathfrak{K}^a  + \frac{1}{\ell^2} V^a h + \frac{1}{2 \ell} \bar{\psi} \Gamma^{a0} \psi \, , \\
R^a & = d V^a + \omega^a_{\phantom{a}b} V^b \, , \\
\mathcal{H} 