\left\{ \tilde{Q}_{\alpha }, \tilde{Q}_{\beta }\right\} = - \frac{1}{2 \ell} \left(\Gamma^{AB} C \right)_{\alpha \beta} \tilde{J}_{AB} + \left( \Gamma ^{A}C\right) _{\alpha \beta }\tilde{P}_{A} \,  ,
\end{split}$$ where $\ell$ is a length parameter, $C$ denotes the charge conjugation matrix, $\Gamma_A$ represents the Dirac matrices in three dimensions (we have $C^T=-C$ and $C \Gamma^A = (C \Gamma^A)^T$), which satisfy the Clifford algebra $\lbrace \Gamma_A, \Gamma_B \rbrace = - 2 \eta_{AB}$, and $\Gamma_{AB} = \frac{1}{2} \left[\Gamma_A, \Gamma_B \right]$.

The non-vanishing components of an invariant tensor for $\mathfrak{osp}(2|1)\otimes \mathfrak{sp}(2)$ are given by $$\begin{split}
& \langle \tilde{J}_{AB} \tilde{J}_{CD} \rangle = \mu_0 \left( \eta_{AD} \eta_{BC} - \eta_{AC} \eta_{BD} \right) \,  , \quad
\langle \tilde{J}_{AB} \tilde{P}_{C} \rangle  = \frac{\mu_1}{\ell} \epsilon_{ABC} \,  , \quad 
\langle \tilde{P}_{A} \tilde{P}_{B} \rangle = \frac{\mu_0}{\ell^2} \eta_{AB} \,  , \\
& \langle \tilde{Q}_\alpha \tilde{Q}_\beta \rangle = \frac{2 \left(\mu_1 -\mu_0 \right)}{\ell} C_{\alpha \beta} \,  ,
\end{split}$$ where $\mu_0$ and $\mu_1$ are arbitrary constants and $\epsilon_{ABC}$ is the Levi-Civita symbol in three dimensions. For convenience, let us redefine the coefficients $\mu_0$ and $\mu_1$ as follows:[^5] $$\mu_0 \rightarrow \alpha_0 \, , \quad \mu_1     \rightarrow \ell \alpha_1 \,  .$$ Thus, the invariant tensor takes the form $$\label{invtosp}
\begin{split}
& \langle \tilde{J}_{AB} \tilde{J}_{CD} \rangle = \alpha_0 \left( \eta_{AD} \eta_{BC} - \eta_{AC} 