of the invariant tensor for the Poincaré superalgebra (the latter arising when the limit $\ell \rightarrow \infty$ is taken in ), and the CS action reduces to $$\label{pcs}
\begin{split}
I^{\text{super-Poincar\'{e}}}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \tilde{\omega}^A_{\phantom{A} B} d \tilde{\omega}^B_{\phantom{B} A} + \frac{2}{3} \tilde{\omega}^A_{\phantom{A} C} \tilde{\omega}^C_{\phantom{C}B} \tilde{\omega}^B_{\phantom{B}A} \right)  + \alpha_1 \left( \epsilon_{ABC} \tilde{R}^{AB} \tilde{V}^C + 2 \bar{\tilde{\psi}} \tilde{D}\tilde{\psi} \right) \\
& - d \left( \frac{\alpha_1}{2} \epsilon_{ABC} \tilde{\omega}^{AB} \tilde{V}^C \right) \Bigg \rbrace \,  ,
\end{split}$$ which is the three-dimensions Poincaré CS supergravity action. Notice that the gravitino does not contribute anymore to the exotic form, which reduces to the Lorentz Lagrangian, while the term proportional to $\alpha_1$ contains the Einstein-Hilbert and Rarita-Schwinger contributions plus a boundary term. Then, omitting the boundary term, we can see that the CS action reproduces the pure three-dimensional supergravity action when the exotic CS term is neglected ($\alpha_0 =0$).

Now, we can pave the way for the development of a three-dimensional CS supergravity theory invariant under the AdS Carroll superalgebra (in $D=3$) introduced in [@Bergshoeff:2015wma]. To this aim, on the same lines of [@Bergshoeff:2015wma], let us decompose the indices as $$\label{indexdec}
A \rightarrow (0,a) \, , \quad a=1,2 \,  .$$ This induces the 