\eta_{BD} \right) \, , \quad
\langle \tilde{J}_{AB} \tilde{P}_{C} \rangle = \alpha_1 \epsilon_{ABC} \,  , \quad
\langle \tilde{P}_{A} \tilde{P}_{B} \rangle = \frac{\alpha_0}{\ell^2} \eta_{AB} \,  , \\
& \langle \tilde{Q}_\alpha \tilde{Q}_\beta \rangle = 2 \left(\alpha_1 -\frac{\alpha_0}{\ell} \right) C_{\alpha \beta} \,  .
\end{split}$$

Then, using the connection $1$-form and the invariant tensor in the general expression , we obtain the $\mathfrak{osp}(2|1)\otimes \mathfrak{sp}(2)$ CS supergravity action, which reads $$\label{CSosp}
\begin{split}
I^{\mathfrak{osp}(2|1)\otimes \mathfrak{sp}(2)}_{CS} & = \frac{k}{4 \pi} \int_\mathcal{M} \Bigg \lbrace \frac{\alpha_0}{2} \left( \tilde{\omega}^A_{\phantom{A} B} d \tilde{\omega}^B_{\phantom{B} A} + \frac{2}{3} \tilde{\omega}^A_{\phantom{A} C} \tilde{\omega}^C_{\phantom{C}B} \tilde{\omega}^B_{\phantom{B}A} + \frac{2}{\ell^2} \tilde{V}^A \tilde{T}_A - \frac{4}{\ell} \bar{\tilde{\psi}} \tilde{\Psi} \right) \\
& + \alpha_1 \left( \epsilon_{ABC} \tilde{R}^{AB} \tilde{V}^C + \frac{1}{3 \ell^2} \epsilon_{ABC} \tilde{V}^A \tilde{V}^B \tilde{V}^C + 2 \bar{\tilde{\psi}} \tilde{\Psi} \right)  - d \left( \frac{\alpha_1}{2} \epsilon_{ABC} \tilde{\omega}^{AB} \tilde{V}^C \right) \Bigg \rbrace \,  ,
\end{split}$$ where the term proportional to $\alpha_0$ is the exotic Lagrangian containing the so-called the Lorentz Lagrangian, a torsional part, and a contribution from the gravitino $1$-form field $\tilde{\psi}$ and its super field-strength. This action describes the most general $\mathcal{N} = 1$, $D = 3$ CS supergravity action (with cosmological constant) for the AdS supergroup [@Giacomini:2006dr].

Notice that in the limit $\ell \rightarrow \infty$ the components in yield the non-vanishing components 