I'm following the notes by Freed about the Dirac Operator. In section 5.4, equation (5.4.25-27), he makes the following claim about the Dirac operator. In a different notation than what he is using, he is considering the operator
$$Q = (\partial_\mu - \frac{1}{8}R_{\beta\mu\nu\sigma} \gamma^\nu \gamma^\sigma x^\beta)^2 = (\partial_\mu - \Omega_{\beta \mu} x^\beta)^2$$
so that $(\Omega_{\beta \mu}) = \frac{1}{8}R_{\beta\mu\nu\sigma} \gamma^\nu \gamma^\sigma$ is an antisymmetric matrix of Clifford algebra elements $\gamma^\mu$, in coordinates $x$ centered around the point of interest. He claims that one can write the matrix $\Omega_{\beta\mu}$ in a block diagonal form
$$\Omega_{\beta\mu} = \begin{pmatrix} 0 &\omega_1 &0 &0 &... &0 &0 \\ -\omega_1 &0 &0 &0 &... &0 &0 \\ 0 &0 &0 &\omega_2 &... &0 &0 \\ 0 &0 &-\omega_2 &0 &... &0 &0 \\ \vdots &\vdots &\vdots &\vdots & &\vdots &\vdots \\ 0 &0 &0 &0 &... &0 &\omega_{d/2} \\ 0 &0 &0 &0 &... &-\omega_{d/2} &0 \\ \end{pmatrix}$$
where the $\omega_i$ are Clifford algebra elements. If one does this, then we can more easily find the heat kernel, since the operator splits up into smaller chunks and use it to show the Dirac operator's index gives the $\hat{A}$ genus.
My question is if such a decomposition is valid, and how one can make sense of these ideas? I've never seen the Riemann tensor decomposed like this and I couldn't find online any resources that indicate this is possible.