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[itex]R^\mu_{\alpha \beta \lambda}[/itex]

when you take covariant derivatives, you get an extra factor of a connection coefficient for each index (with a minus sign in the case of lowered indices). So:

[itex]\nabla_\nu R^\mu_{\alpha \beta \lambda} = \partial_\nu R^\mu_{\alpha \beta \lambda} + \Gamma^\mu_{\nu \sigma} R^\sigma_{\alpha \beta \lambda} - \Gamma^\tau_{\nu \alpha} R^\mu_{\tau \beta \lambda} - \Gamma^\tau_{\nu \beta} R^\mu_{\alpha \tau \lambda} - \Gamma^\tau_{\nu \lambda} R^\mu_{\alpha \beta \tau}[/itex]

What else are you wanting to know about it? Did you want to write out [itex]R[/itex] in terms of [itex]\Gamma[/itex]? That's an enormous mess.

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[itex]R^\mu_{\alpha \beta \lambda}[/itex]

when you take covariant derivatives, you get an extra factor of a connection coefficient for each index (with a minus sign in the case of lowered indices). So:

[itex]\nabla_\nu R^\mu_{\alpha \beta \lambda} = \partial_\nu R^\mu_{\alpha \beta \lambda} + \Gamma^\mu_{\nu \sigma} R^\sigma_{\alpha \beta \lambda} - \Gamma^\tau_{\nu \alpha} R^\mu_{\tau \beta \lambda} - \Gamma^\tau_{\nu \beta} R^\mu_{\alpha \tau \lambda} - \Gamma^\tau_{\nu \lambda} R^\mu_{\alpha \beta \tau}[/itex]

What else are you wanting to know about it? Did you want to write out [itex]R[/itex] in terms of [itex]\Gamma[/itex]? That's an enormous mess.

I started with what you had written down in your post, but as you said it is a mess, and I am mixing up something due to enormous mess. I just want to see the full solution and I do not want you to be exhausted to derive the solution on this forum, so that is why I am asking you to share a file or other things which include this enormous solution) If you have, I will be very pleased

[itex]R^\mu_{\alpha \beta \lambda}[/itex]

when you take covariant derivatives, you get an extra factor of a connection coefficient for each index (with a minus sign in the case of lowered indices). So:

[itex]\nabla_\nu R^\mu_{\alpha \beta \lambda} = \partial_\nu R^\mu_{\alpha \beta \lambda} + \Gamma^\mu_{\nu \sigma} R^\sigma_{\alpha \beta \lambda} - \Gamma^\tau_{\nu \alpha} R^\mu_{\tau \beta \lambda} - \Gamma^\tau_{\nu \beta} R^\mu_{\alpha \tau \lambda} - \Gamma^\tau_{\nu \lambda} R^\mu_{\alpha \beta \tau}[/itex]

What else are you wanting to know about it? Did you want to write out [itex]R[/itex] in terms of [itex]\Gamma[/itex]? That's an enormous mess.

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It's a useless exercise if you ask me.

Other than that you could use mathematica but it will still take a bit of writing, especially paying attention that you have a valid expression.

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