Spin connection of a conformally flat metric (${\omega }_{a\mu \nu }={\eta }_{a\mu }{\partial }_{\nu }\log \Omega -{\eta }_{a\nu }{\partial }_{\mu }\log \Omega$)

For $g_{ab}={\Omega }^2{\eta }_{ab}$, use the vierbein $e^{\mu }{}_a=\Omega {\delta }^{\mu }{}_a$ and constant flat gamma matrices. The tetrad postulate and ${\Gamma }^c{}_{ab}={\delta }_a^c{\partial }_b\log \Omega +{\delta }_b^c{\partial }_a\log \Omega -{\eta }_{ab}{\eta }^{cd}{\partial }_d\log \Omega$ give the displayed lowered spin connection. With $L={\hat{\gamma }}^c{\partial }_c\log \Omega$, its spinor action is ${\nabla }_a={\partial }_a+[{\hat{\gamma }}_a,L]/4$. The Lorentz index lowering matters in indefinite signature.