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I have to compute the square of the Dirac operator, $D=\gamma^a e^\mu_a D_\mu$ , in curved space time ($D_\mu\Psi=\partial_\mu \Psi + A_\mu ^{ab}\Sigma_{ab}$ is the covariant derivative of the spinor field and $\Sigma_{ab}$ the Lorentz generators involving gamma matrices). Dirac equation for the massless fermion is $\gamma^a e^\mu_a D_\mu \Psi=0$. In particular I have to show that Dirac spinors obey the following equation: $$(-D_\mu D^\mu + \frac{1}{4}R)\Psi=0 \qquad (1)$$ where R is (I guess) the Ricci scalar. Appling to the Dirac eq, the operator $\gamma^\nu D_\nu$ and decomposing the product $\gamma^\mu \gamma^\nu$ in symmetric and antisymmetric part I found: $$D_\mu D^\mu\Psi + \frac{1}{4}[\gamma^\mu,\gamma^\nu][D_\mu, D_\nu]\Psi=0$$Now I have troubles to show that this last object is related with the Ricci scalar. Can somebody help me or suggest me the right way to solve Eq. $(1)$?

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