General relativity. Bianchi identities

[TheSource007]

Homework Statement

I have a problem. I need to prove that the divergence of Einstein tensor is 0 using the bianchi identities. I have looked to several sources and I have derived an answer, but I don't fully understand some steps.

Homework Equations

I have uploaded a document which shows a part of the solution, because I have no idea hot to write them in the post. The solution starts with eq0.
I don't understand how did they get from eq 2 to eq 3. It seems like some sort of contraction in the index a and u, but what kind or rule/law/equation is that? If it's a contraction between first and third slots, why didnt they contract also the third term in eq 2?
My second question is, when raising an index, how the metric in equation 5 is distributed to get eq6? It seems that the metric is distributed like simple algebra. Why is it like that? How the covariant derivative affect the index's raising.
If you can help me with those two questions, then I will be able to understand the rest.
Please be kind since I am very new at this stuff. The book I am using is gravitation by misner thorne wheeler.

The Attempt at a Solution

 

[member 553083]

I have attempted to answer the questions by looking at various sources. However, I am still not sure how to answer them.