Simplification of Odd Parity Curvature Squared Terms

[sugr-wq]

I need Mathematica to simplify a couple of odd-parity curvature-squared terms, using all the symmetry properties and identities of the Riemann tensor and the Ricci tensor along with dimensional-dependent identities in 4D spacetime.

The terms are
$$ \epsilon_{\beta \gamma\delta\epsilon}, R^{\alpha \beta}, R_\alpha{}^{\gamma \delta \epsilon},\  \epsilon_{\alpha \beta \gamma \delta}, R, R^{\alpha \beta \gamma \delta},\ \epsilon_{\gamma \delta \epsilon \zeta}, R^{\alpha \beta \gamma \delta}, R_{\alpha \beta}{}^{ \epsilon \zeta},\ \epsilon_{\beta \delta \epsilon\zeta}, R^{\alpha \beta \gamma \delta}, R_{\alpha \gamma}{}^{\epsilon \zeta},\ \epsilon_{\beta\delta\epsilon\zeta}, R^{\alpha \beta \gamma \delta}, R_\alpha{}^\epsilon{}_\gamma{}^\zeta$$

Out of the five terms above, $\epsilon_{\gamma \delta \epsilon \zeta}$, $R^{\alpha \beta \gamma \delta}$, $R_{\alpha \beta}{}^{ \epsilon \zeta}$ is the only one that is non-zero. Rest of the terms are supposed to vanish on simplification.

I was able to show that the second term, i.e. $\epsilon_{\alpha \beta \gamma \delta}, R, R^{\alpha \beta \gamma \delta}$, vanishes, by using the command RiemannToChristoffel followed by ToCanonical. However this command does not show the vanishing of the other terms.

Any suggestions for how to show that the remaining curvature squared terms which are supposed to vanish, vanish?