# Calculations with tensors give two different results from seemingly equivalent paths:

$$\require{cancel}$$ I want to calculate some operators in the context of Metric-Affine-Gravity: specifically spinors. I will work in tangent space, where all greek ($$\mu,\nu,..$$) indices are cast into latin ones ($$a,b,..$$) via use of tetrads (for tensors) and with the appropriate rules for other indexed quantities (like connections).
Define the covariant derivative$$\nabla_c=\partial_c+\tfrac{1}{2}\omega_{abc}\sigma^{ab}$$ with $$\sigma^{ab}=\tfrac{1}{4}[\gamma^a,\gamma^b]\equiv\tfrac{1}{2}\gamma^{cd}$$ so that $$\omega_{abc}={\overset{\phantom{.}\circ}{\omega}}_{abc}+\Omega_{abc}$$ is a general spin connection on spinors assumed to be metric compatible only, and thus has torsion. Here $${\overset{\phantom{.}\circ}{\omega}}$$ and $$\Omega$$ are the tangent space equivalents of christoffel symbols and contortion respectively (see Wikipedia's page on contorsion for the spacetime indexed versions of them). So we can split $$\nabla=\overset{\circ}{\nabla}+\tfrac{1}{4}\Omega$$
I want to calculate the following term (considering $$\gamma^c$$ are covariantly conserved) \begin{align} \begin{split} {\cancel{\nabla}}{\cancel{\nabla}}&=\gamma^c\gamma^d\nabla_c\nabla_d=\Big( \tfrac{1}{2}\{\gamma^c,\gamma^d\} +\tfrac{1}{2}[\gamma^c,\gamma^d] \Big)\nabla_c\nabla_d=\\ &=\nabla_c\nabla^c+\tfrac{1}{2}\gamma^{cd}[\nabla_c,\nabla_d] \end{split} \end{align} now, $$\phantom{cd}\nabla_c\nabla^c={\overset{\circ}{\nabla}}_c{\overset{\circ}{\nabla}}\vphantom{A}^c+\tfrac{1}{4}({\overset{\circ}{\nabla}}_c\Omega^c+\Omega^c{\overset{\circ}{\nabla}}_c)+\tfrac{1}{16}\Omega_c\Omega^c$$
The curvature 2-form is: $$\phantom{b} R(X,Y)Z=[\nabla_X,\nabla_Y]Z-\nabla_{[X,Y]}Z\phantom{cb}$$, so when applied to spinors, this becomes the commutator of covariant derivatives $$\nabla_c$$ minus the torsion part $$\phantom{cb}{\Omega_{ab}}^c-{\Omega_{ba}}^c={T^a}_{bc}$$ coming from $$\Omega$$ : so $$\phantom{b}\tfrac{1}{2}\gamma^{cd}[\nabla_c,\nabla_d]=\tfrac{1}{8}\gamma^{cd}R_{abcd}\gamma^{ab}+\tfrac{1}{8}\gamma^{ab}{T^c}_{ab}\nabla_c \phantom{b}$$
$$\phantom{b}R_{abcd}=\partial_c\omega_{abd}+\omega_{axc}{\omega^x}_{bd}-c\leftrightarrow d$$. Substituting $$\omega={\overset{\phantom{.}\circ}{\omega}}+\Omega$$ and calling $$\Omega_{abc}\gamma^{ab}\equiv\Omega_c$$,
$$\phantom{cb} \tfrac{1}{8}\gamma^{cd}R_{abcd}\gamma^{ab}=\tfrac{1}{8}\gamma^{cd}\overset{\circ}{R}_{abcd}\gamma^{ab}+\tfrac{1}{2}\gamma^{cd}{\overset{\circ}{\nabla}}_c\Omega_d+\tfrac{1}{8}\gamma^{cd}\Omega_{axc}{\Omega^x}_{bd}\gamma^{ab}$$
Had we substituted $$\nabla={\overset{\circ}{\nabla}}+\tfrac{1}{4}\Omega$$ in the first equation, we would have gotten \begin{align} \begin{split} (\overset{\circ}{\cancel{\nabla}}+\tfrac{1}{4}\cancel{\Omega})(\overset{\circ}{\cancel{\nabla}}+\tfrac{1}{4}\cancel{\Omega})&=\overset{\circ}{\cancel{\nabla}}\cancel{\nabla}+\tfrac{1}{4}(\overset{\circ}{\cancel{\nabla}}\cancel{\Omega}+\cancel{\Omega}\overset{\circ}{\cancel{\nabla}})+\tfrac{1}{16}\cancel{\Omega}\cancel{\Omega}=\\ &={\overset{\circ}{\nabla}}_c{\overset{\circ}{\nabla}}^c+\tfrac{1}{2}\gamma^{cd}[{\overset{\circ}{\nabla}}_c,{\overset{\circ}{\nabla}}_d]+\tfrac{1}{16}\Omega_c\Omega^c\\ &+\tfrac{1}{4}({\overset{\circ}{\nabla}}_c\Omega^c+\Omega^c{\overset{\circ}{\nabla}}_c+\gamma^{cd}({\overset{\circ}{\nabla}}_c\Omega_d+\Omega_c{\overset{\circ}{\nabla}}_d)) \end{split} \end{align} This time, $$\phantom{cd}\tfrac{1}{2}\gamma^{cd}[{\overset{\circ}{\nabla}}_c,{\overset{\circ}{\nabla}}_d]$$ is simply $$\tfrac{1}{8}\gamma^{cd}\overset{\circ}{R}_{abcd}\gamma^{ab}$$ because $${\overset{\circ}{\nabla}}$$ has no torsion. In the end, we have
\begin{align} \begin{split} {\cancel{\nabla}}{\cancel{\nabla}}&={\overset{\circ}{\nabla}}_c{\overset{\circ}{\nabla}}\vphantom{A}^c+\tfrac{1}{4}({\overset{\circ}{\nabla}}_c\Omega^c+\Omega^c{\overset{\circ}{\nabla}}_c)+\tfrac{1}{16}\Omega_c\Omega^c +\\ &+\tfrac{1}{8}\gamma^{cd}\overset{\circ}{R}_{abcd}\gamma^{ab}+\tfrac{1}{4}\gamma^{cd}{\overset{\circ}{\nabla}}_c\Omega_d+\\ &+\tfrac{1}{16}\gamma^{cd}\Omega_{axc}{\Omega^x}_{bd}\gamma^{ab}+\tfrac{1}{4}\Omega^c\nabla_c \phantom{b} \\ \neq(\overset{\circ}{\cancel{\nabla}}+\tfrac{1}{4}\cancel{\Omega})(\overset{\circ}{\cancel{\nabla}}+\tfrac{1}{4}\cancel{\Omega})&={\overset{\circ}{\nabla}}_c{\overset{\circ}{\nabla}}^c+\tfrac{1}{4}({\overset{\circ}{\nabla}}_c\Omega^c+\Omega^c{\overset{\circ}{\nabla}}_c)+\tfrac{1}{16}\Omega_c\Omega^c+\\ &+\tfrac{1}{8}\gamma^{cd}\overset{\circ}{R}_{abcd}\gamma^{ab}+\tfrac{1}{4}\gamma^{cd}({\overset{\circ}{\nabla}}_c\Omega_d+\Omega_c{\overset{\circ}{\nabla}}_d) \end{split} \end{align} so, what have I not understood, or what am I doing wrong? Algebraically this all seems to make sense to me.
One error could be the extra torsion term (last term in the first expansion) from the commutator of $$\nabla$$s, but that would still not account for the extra $$\gamma^{cd} \nabla_c \Omega_d$$ and $$\gamma\Omega\Omega\gamma$$ terms