$F$ in , $\delta_\varrho F = \left[ F, \varrho \right] $, we can write the gauge transformations $$\label{curvtr}
\begin{split}
\delta \mathcal{R}^{ab} & = \frac{2}{\ell^2} R^a \varrho^b \,  , \\
\delta \mathcal{K}^a  & = \mathcal{R}^a_{\phantom{a} b} \kappa^b - \varrho^a_{\phantom{a} b} \mathcal{K}^b - \frac{1}{\ell^2} \varrho^a \mathcal{H} + \frac{1}{\ell^2} R^a \tau - \frac{1}{\ell} \bar{\varepsilon} \Gamma^{a0} \Psi \,  , \\
\delta R^a & = \mathcal{R}^a_{\phantom{a} b} \varrho^b - \varrho^a_{\phantom{a} b} R^b \,  , \\
\delta \mathcal{H} & = - \varrho^a \mathcal{K}_a + R^a \kappa_a + \bar{\varepsilon} \Gamma^0 \Psi \, , \\
\delta \Psi & = \frac{1}{4} \mathcal{R}^{ab} \Gamma_{ab} \varepsilon - \frac{1}{4} \varrho^{ab} \Gamma_{ab} \Psi - \frac{1}{2 \ell} \varrho^a \Gamma_a \Psi + \frac{1}{2 \ell} R^a \Gamma_a \varepsilon \, .
\end{split}$$

The equations of motion derived from the variation of the action with respect to the fields $\omega^{ab}$, $k^a$, $V^a$, $h$, and $\psi$ are $$\label{eom}
\begin{split}
& \delta \omega^{ab} \; : \quad \alpha_0 \mathcal{R}^{ab} + \alpha_1 \epsilon^{ab} \mathcal{H}= 0 \, , \quad \quad \quad
\delta k^a \; : \quad  \alpha_1 R^a = 0 \, , \\
& \delta V^a \; : \quad \frac{\alpha_0}{\ell^2} R^a + 2 \alpha_1 \epsilon_{ab} \mathcal{K}^b = 0 \,  , \quad \quad \quad 
\delta h \; : \quad \alpha_1 \mathcal{R}^{ab} = 0  \, 