Dirac equation for the massless fermions in curved spase time is $γ^ae^μ_aD_μΨ=0$, where $e^μ_a$ are the tetrads. I have to show that Dirac spinors obey the following equation: $$(−D_μD^μ+\frac{1}{4}R)Ψ=0\qquad(1)$$
where $R$ is the Ricci scalar.
I already know that $[D_\mu,D_\nu]A^\rho={{R_{\mu\nu}}^\rho}_\sigma A^\sigma$, but a key point is to know what $[D_\mu,D_\nu]\Psi$ is.
($D_μΨ=∂_μΨ+A^{ab}_μΣ_{ab}$ is the covariant derivative of the spinor field and $Σ_{ab}$ the Lorentz generators involving gamma matrices).