How did the author do the cyclic permutation?
$\Gamma^k_{ij}g_{kl}+\Gamma^k_{lj}g_{ki}=\partial_jg_{il}$
We can cyclically permute these indices to generate two more equations:
$\Gamma^k_{jl}g_{ki}+\Gamma^k_{il}g_{kj}=\partial_lg_{ji}$
$\Gamma^k_{li}g_{kj}+\Gamma^k_{ji}g_{kl}=\partial_ig_{lj}$
Source: http://www.physicspages.com/2013/12/22/christoffel-symbols-in-terms-of-the-metric-tensor/
Isn't it supposed to be one cycle for all of them in each equation like those shown below? $\Gamma^k_{jl}g_{ki}+\Gamma^k_{ji}g_{kl}=\partial_ig_{lj}$
$\Gamma^k_{li}g_{kj}+\Gamma^k_{il}g_{kj}=\partial_lg_{ji}$
Thanks for all the help