The most useful reference is ch. 13 of GR by Wald. But in book there are some mistakes in numerical coefficients.
Expansion of Rimannian thensor:
\begin{equation} R_{\alpha\dot{\alpha}\beta\dot{\beta}\gamma\dot{\gamma}\delta\dot{\delta}} = \Psi_{( \alpha\beta\gamma\delta)} \epsilon_{\dot{\alpha}\dot{\beta}} \epsilon_{\dot{\gamma}\dot{\delta}} + \Phi_{(\dot{\alpha}\dot{\beta})(\gamma\delta)} \epsilon_{\alpha\beta} \epsilon_{\dot{\gamma}\dot{\delta}} + \Lambda (\epsilon_{\alpha\gamma}\epsilon_{\beta\delta} + \epsilon_{\beta\gamma}\epsilon_{\alpha\delta}) \epsilon_{\dot{\alpha}\dot{\beta}} \epsilon_{\dot{\gamma}\dot{\delta}} + c.c. \end{equation}
\begin{equation} 16 \, R_{abcd} = - \Psi_{( \alpha\beta\gamma\delta)} (\sigma_{ab})^{\alpha\beta} (\sigma_{cd})^{\gamma\delta} + \Phi_{(\dot{\alpha}\dot{\beta})(\gamma\delta)} (\sigma_{ab})^{\alpha\beta} (\tilde{\sigma}_{cd})^{\dot{\gamma}\dot{\delta}} -4 \Lambda (\eta_{a d} \eta_{bc} - \eta_{ac} \eta_{bd} - i \epsilon_{abcd}) + c.c. \end{equation}
$$ 16 R = -4 \Lambda g^{ac}g^{bd} (\eta_{a d} \eta_{bc} - \eta_{ac} \eta_{bd}) = -4 \Lambda (4-16) = 48 \Lambda \;\;\; \Rightarrow \;\;\; \Lambda = \frac{R}{3} $$
\begin{equation} \sigma^{ab}_{\alpha\beta} \sigma^{cd}_{\gamma\delta} R_{abcd} = - \Psi_{( \alpha\beta\gamma\delta)} + 4 \Lambda \epsilon_{(\alpha\gamma} \epsilon_{\beta)\delta} + c.c. \end{equation}