\tilde{Q}_{\alpha }\right] = -\frac{1}{2 \ell} \left( \Gamma _{a}\tilde{Q}\right) _{\alpha } \, , \quad \quad 
\left[ \tilde{H}, \tilde{Q}_{\alpha }\right] = -\frac{1}{2 \ell} \left( \Gamma _{0}\tilde{Q}\right) _{\alpha } \,  , \\
& \left\{ \tilde{Q}_{\alpha }, \tilde{Q}_{\beta }\right\} = - \frac{1}{2 \ell} \left(\Gamma^{ab} C \right)_{\alpha \beta} \tilde{J}_{ab} - \frac{1}{\ell} \left(\Gamma^{a0} C \right)_{\alpha \beta} \tilde{K}_{a}  + \left( \Gamma ^{a}C\right) _{\alpha \beta }\tilde{P}_{a} + \left( \Gamma ^{0}C\right) _{\alpha \beta }\tilde{H} \,  .
\end{split}$$

Moreover, the connection $1$-form and the curvature $2$-form can be respectively rewritten as $$\label{conndec}
\tilde{A} = \frac{1}{2} \tilde{\omega}^{ab} \tilde{J}_{ab} + \tilde{k}^a \tilde{K}_a + \tilde{V}^a \tilde{P}_a + \tilde{h} \tilde{H} + \tilde{\psi}^\alpha \tilde{Q}_\alpha \,  , \quad
\tilde{F} = \frac{1}{2} \tilde{\mathcal{R}}^{ab} \tilde{J}_{ab} + \tilde{\mathcal{K}}^a \tilde{K}_a + \tilde{R}^a \tilde{P}_a + \tilde{\mathcal{H}} \tilde{H} + \tilde{\Psi}^\alpha \tilde{Q}_\alpha \,  ,$$ with $$\label{curvdec}
\begin{split}
\tilde{\mathcal{R}}^{ab} & = d \tilde{\omega}^{ab} - \tilde{k}^a \tilde{k}^b + \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{\mathcal{K}}^a  & = d \tilde{k}^a + \tilde{\omega}^a_{\phantom{a} b} \tilde{k}^b + \frac{1}{\ell^2} \tilde{V}^a \tilde{h} + \frac{1}{2 \ell} \bar{\tilde{\psi}} \Gamma^{a0} \tilde{\psi} = \tilde{\mathfrak{K}}^a  + \frac{1}{\ell^2} \tilde{V}^a \tilde{h} + \frac{1}{2 \ell} \bar{\tilde{\psi}} \Gamma^{a0} \tilde{\psi} \, , \\
\tilde{R}^a & = d \tilde{V}^a + \tilde{\omega}^a_{\phantom{a}b} \tilde{V}^b + \tilde{k}^a \tilde{h} 