There is the usual expression for the Riemann tensor
$$R_{abcd}=\partial_c\Gamma_{adb}-\partial_d\Gamma_{acb}+\Gamma_{ace}{\Gamma^e}_{db}-\Gamma_{ade}{\Gamma^e}_{cb}.$$ However, in the last page of https://www.mathi.uni-heidelberg.de/~walcher/teaching/wise1516/geo_phys/SigmaAndLGModels.pdf, another expression is used
$$R_{abcd}=\partial_c\Gamma_{adb}-\partial_d\Gamma_{acb}+\Gamma_{ead}{\Gamma^e}_{cb}-\Gamma_{eac}{\Gamma^e}_{db}.$$
How does one obtain the first expression from the second? I've never seen the second expression before. The first one is obvious from the expression $R=\text{d}\Gamma-[\Gamma\wedge\Gamma]$. However, the second one is less intuitive. In orthogonal coordinates it would be easy to obtain since $$\Gamma_{abc}=-\Gamma_{bac}.$$ However, is there a way to see this two expression are equivalent without using orthogonality? I think it will have to do with the metricity condition $$\partial_ag_{bc}=\Gamma_{bca}-\Gamma_{cba}.$$
Bonus question: How do you do index placement in stackexchange?
The metric tensor $g_{fe}$ is used to lower the index $f$ and rename it to $e$: $$ g_{fe}\Gamma^f_{md}=\Gamma_{emd}\,. $$ and $g^{am}$ is used to raise $m$ and rename it to $a$. Try to apply this to the expression $\Gamma^f_{md}\Gamma^e_{cb}-\Gamma^f_{mc}\Gamma^e_{db}\,.$ It must also have some symmetry properties which should finally give the first expression.
Index placement I always to like this:
${R^a}_{bcd}$