Can the Ricci Tensor be Simplified Further? Suggestions Needed!

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In my studies of methods to simplify the Einstein field equations, I first decided to go about expanding the Ricci tensor in terms of the metric tensor. I have been mostly successful in doing this, but there are a couple of complications that I would like your opinions on.

At the bottom of this post, I will leave an attachment that leads to the word document that shows my work when expanding the Ricci tensor. In this attachment, I have highlighted two sections. Now here are my questions:

1. In my first highlighted section, is there any way to further simplify this expression for the derivative of this Christoffel symbol? Is there perhaps some property of Christoffel symbols that I can use?

2. In my second highlighted section at the bottom, how would I express the multiplication of those two Christoffel symbols without overloading any indices (especially the c index)? Can somebody please type out that expression so that I can see what I should do when multiplying two Christoffel symbol formulas?

3. Does everything else look right to you all? Is there any way to simplify any other part that I did not ask about?

Here is the the attachment:

View attachment typing_the_terms_of_the_ricci_tensor 2._docx.docx

P.S. I would have just posted the attachment here directly on the forum, but I could not do this for some unknown reason when I clicked on the attachment button.

 

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Everyone, I got on another computer at school and this computer allowed me to post an attachment. Hopefully now you all will have easier access to the document and be able to reply. I put the attachment at the bottom of the OP.