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I was reading a book about general relativity and I came across these two equations

$$ \begin{align}

\mathrm{g}^{\mu\nu}_{,\rho}+

\mathrm{g}^{\sigma\nu}{\Gamma}^{\mu}_{\sigma\rho}+

\mathrm{g}^{\mu\sigma}{\Gamma}^{\nu}_{\rho\sigma}

-\frac{1}{2}(

{\Gamma}^{\sigma}_{\sigma}+{{\Gamma}^{\sigma}_{\sigma\rho}}

)

\mathrm{g}^{\mu\nu}

&=0,

\tag1

\\

\mathrm{g}^{[\mu\nu]}_{,\nu}

-\frac{1}{2}(

{\Gamma}^{\rho}_{\rho\nu}+{\Gamma}^{\rho}_{\nu\rho}

)

\mathrm{g}^{(\mu\nu)}

&=0,

\tag2

\end{align}

$$

and it says that by contracting equation (1) once with respect to ($\mu,\rho$), then with respect to ($\nu,\rho$), then subtracting the resulting equations we can get equation (2), however I can't see how that's possible.

Also, This connection is not symmetric with respect to the lower Indices

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# Contracting Indices

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