I am trying to get an explicit expression for the variation $\delta \omega_{\mu}^{\ ab} / \delta e_\mu^a$, but when doing the actual variation I end up with a series of 16 terms that I cannot simplify any further, though it should be possible to rewrite the resulting expression as a sum of covariant derivatives of the vielbein. The resulting identity is relevant for several calculations that involve spinors on curved spacetimes and Supergravity, but I cannot find a similar calculation or result anywhere for reference.
Edit: If someone knows a good Mathematica package to take variational derivatives of the vielbein and spin connection, that would also be very helpful.
Requiring the spin connection to be torsion free and compatible with the metric gives us the following constraint $$ \nabla_\mu e_\nu^a = \partial_\mu e_\nu^a + \omega_{\mu\ b}^{\ a} e^b_\nu - \Gamma^{\ \lambda}_{\mu\ \nu} e^a_\lambda = 0 $$ This allows us to write the following expression for the spin connection in terms of the vielbein $$ \begin{align} \omega_{\mu}^{\ ab} =& e^{\nu a} \Gamma^\lambda_{\ \mu\nu} e^b_\lambda - e^{\nu a} \partial_\mu e_\nu ^{\ b} \\ =& \frac{1}{2} e^{\nu a} g^{\lambda\sigma} \left( \partial_\mu g_{\sigma\nu} + \partial_\sigma g_{\nu\mu} - \partial_\nu g_{\mu\sigma} \right) - e^{\nu a} \partial_\mu e_\nu ^{\ b} \\ =& \frac{1}{2}e^{\nu a}(\partial_\mu e_\nu^{\ b}-\partial_\nu e_\mu^{\ b})-\frac{1}{2}e^{\nu b}(\partial_\mu e_\nu^{\ a}-\partial_\nu e_\mu^{\ a})-\frac{1}{2}e^{\rho a}e^{\sigma b}(\partial_\rho e_{\sigma d}-\partial_\sigma e_{\rho d})e_\mu^{\ d} \\ =& \frac{1}{2} e^{\nu [a} \left( \partial_{\mu} e_{\nu}^{b]} - \partial_{\nu} e_{\mu}^{b]} + e^{b]\sigma}e_\mu^d\partial_\sigma e_ {\nu d} \right) \end{align} $$ where I first wrote the Christoffel connection in terms of the metric and then used that $g_{\mu\nu} = e^a_\mu e^b_\nu \eta_{ab}$.
Does someone know a source where there is an expression for the following variational derivative? $$ \frac{\delta \omega_{\mu}^{\ ab}}{\delta e_\mu^a} $$ I tried calculating it myself, but since the above expression has four bilinear terms in $e$ and two quartic terms in $e$, you will end up with $2 \times 4 + 4 \times 2 = 16$ terms after doing the actual variation. I have a lot of trouble rewriting this variation as a neat expression (and cannot find any references that mention it)
As a comparison, consider the variation of the Christoffel connection with respect to the metric. $$ \Gamma^\mu_{\nu\rho} = g^{\mu\lambda} \Gamma_{\lambda\nu\rho} = \frac{1}{2} g^{\mu\lambda} \left( \partial_\lambda g_{\nu\rho} + \partial_\nu g_{\rho\lambda} - \partial_\rho g_{\lambda\nu} \right) $$ When varying this expression you will also get a lengthy series of terms, but the trick is to see they can be rearanged as covariant derivatives of the Christoffel connection. I assume there must be a similar trick and expression to rewrite the variation of the spin connection in terms of covariant derivatives of the vielbein? $$ \begin{align} \delta \Gamma^\mu_{\nu\rho} =& \Gamma_{\lambda\nu\rho} \delta g^{\mu\lambda} + g^{\mu\lambda} \delta \Gamma_{\lambda\nu\rho} \\ =& \frac{1}{2}g^{\mu\lambda} \left( \nabla_\lambda \delta g_{\nu\rho} + \nabla_\nu \delta g_{\rho\lambda} - \nabla_\rho \delta g_{\lambda\nu} \right) \end{align} $$
