Quick question, how do I compute the last 4 Christoffel symbols for the second type of Riemann Tensor (2nd equation in the picture)? I am having a problem understanding how you can compute those symbols with them having an additional index to that of the Riemann tensor you're trying to solve for. I'm fairly new to tensors so this is probably a dumb question in general but any help is welcomed.

This is a link to a paper to get an idea of my question: https://physicspages.com/pdf/Moore/Moore%20Problems%2018.06.pdf

Equations 10 and 12 on page 2; they seem to substitute the free indice with whatever makes the symbol not go to 0, but you also have other symbols that don't go to zero, so is every different combination of substitutions for that indice another solution to the tensor?

Thanks for your help!