Ricci curvature under Ricci flow In fact ,I got stuck in some compute. Let $B_{ijkl}=g^{pr}g^{qs}R_{piqj}R_{rksl}$,
in the below picture,I can't compute out the 1 and 2 equation above red line.
In the first, I know how to get $\Delta R_{ik}$, but the other part ,it's seemly too complex that I can't compute out.
In the second , I think I need some exchange equation of $B_{ijkl}$.
Besides, I always get stuck in compute , when I read paper about Ricci flow.
What should I do? And there are so many index that let me dazzle, whether there are some easy way to deal it ? 
Thanks for useful hint or answer.Sorry for my poor English.
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 A: (1) If we use normal coordinate, then the term in second line is
$$
-g^{jl}g^{pq} (R_{pjkl} R_{qi} + R_{ipkl} R_{qj} + R_{ijpl} R_{qk} +
R_{ijkp} R_{ql} ) $$
$$ = -g^{pq} ( R_{pk} R_{qi} + R_{ipkl} R_{ql} +
R_{ip} R_{qk} + R_{ilkp} R_{ql} ) $$
$$ =-2g^{pq} R_{pk} R_{qi}\underbrace{ - g^{pq} (  R_{ipkl} R_{ql} +
 R_{ilkp} R_{ql} )}_{=A} $$
The term in the first line is $$ 2g^{jl} ( B_{ijkl} - B_{ijlk} -
B_{il jk} + B_{ikjl} ) = 2g^{jl} ( B_{ijkl} - 2B_{ijlk} )
 + \underbrace{2B_{ikll} }_{=B} $$
so that $$ A+ B = - g^{pq} (  R_{ipkl} R_{ql} +
 R_{ilkp} R_{ql} )
 +2 R_{piqk } R_{pq} =0 $$
by Bianchi identity
(2) Recall $$ B_{ijkl} = R_{piqj} R^{p\ q}_{\ k\ l} $$
 so that $$ g^{jl} B_{ijkl}= R_{piqj} R^{p\ qj}_{\ k} = R_{piqj} R^{qjp}_k $$
 $$ g^{jl} B_{ijlk}= R_{piqj} R^{pj q}_{\ \ \ k} $$
The term in third line can be expanded into $$
 R_{piqj} R^{p\ qj}_{\ k}  - R_{pjqi} R^{p\ qj}_{\ k} - R_{piqj} R^{pjq}_{\ \ \ k} +
 R_{pjqi } R^{pjq}_{\ \ \ k}$$
$$ =
 R_{piqj} R^{qjp}_{\ \ \ k}  - R_{pjqi} R^{qjp}_{\ \ \ k} -
 g^{jl} B_{ijlk}  +
 R_{qjpi } R^{qjp}_{\ \ \ k}$$
$$ = g^{jl} B_{ijkl} -  g^{jl} B_{ijlk}-
 g^{jl} B_{ijlk}  +
g^{jl} B_{ijkl}$$ Hence we complete the proof
