Finally! Since Majorana spinors satisfy
\begin{align}
\overline{\psi} \rho_a \psi = - \overline{\psi} \rho_a \psi = 0
\end{align}
we see, using the spinor covariant derivative
\begin{align}
\nabla_a &= \partial_a + \frac{1}{4}\omega_a^{\mu \nu} \Gamma_{\mu \nu} \\
&= \partial_a + \frac{1}{8}\omega_a^{\mu \nu}[\rho_{\mu},\rho_{\nu}],
\end{align}
and the Clifford relations
\begin{align}
\{ \rho^a,\rho^b \} &= - 2 \eta^{ab}, \eta^{ab} = (-1,1), \\
\{\rho^0,\rho^0 \} &= - 2 \eta^{00} \\
2(\rho^0)^2 &= - 2 (-1) \\