Contracting Indices Does anyone know how to get from (1) to (2) in the system
$$    \begin{align}
    \mathrm{g}^{\mu\nu}_{,\rho}+
    \mathrm{g}^{\sigma\nu}{{\Gamma}}^{\mu}_{\sigma\rho}+
    \mathrm{g}^{\mu\sigma}{{\Gamma}}^{\nu}_{\rho\sigma}
    -\frac{1}{2}(
    {{\Gamma}}^{\sigma}_{\rho\sigma}+{{\Gamma}}^{\sigma}_{\sigma\rho}
    )
    \mathrm{g}^{\mu\nu}
    &=0,
\tag1
  \\
    \mathrm{g}^{[\mu\nu]}_{,\nu}
    +\frac{1}{2}(
    {{\Gamma}}^{\rho}_{\rho\nu}-{{\Gamma}}^{\rho}_{\nu\rho}
    )
    \mathrm{g}^{(\mu\nu)}
    &=0,
\tag2
    \end{align}
$$
by contracting equation (1) once with respect to ($\mu,\rho$), then with respect to ($\nu,\rho$)?
Where $\Gamma$ is not symmetric with respect to the lower Indices.
My attempt so far to solve this problem is:
Well, when contracting with respect to μ and ρ I get:
$$
-\frac{1}{2} g^{\rho \nu } {\Gamma} _{a\rho}^{a}-\frac{1}{2} g^{\rho
   \nu} \Gamma _{\rho a}^{a}+g^{a\nu} \Gamma _{a\rho}^{\rho}+g^{\rho a} \Gamma _{\rho a}^{\nu}+g_{,\rho}^{\rho\nu}=0
$$
and when contracting with respect to nu and ρ I get:
$$
-\frac{1}{2} g^{\mu \rho } \Gamma _{a\rho}^a-\frac{1}{2} g^{\mu
   \rho} \Gamma _{\rho a}^a+g_{}^{a\rho } \Gamma _{a\rho}^{\mu }+g^{\mu a} \Gamma _{\rho a}^{\rho
   }+g_{,\rho }^{\mu \rho }=0
$$
when subtracting these two equations I get:
$$
\frac{1}{2} g_{}^{\mu \rho } \Gamma _{a\rho }^a-\frac{1}{2} g_{}^{\rho
   \nu } \Gamma _{a\rho }^a+\frac{1}{2} g_{}^{\mu \rho } \Gamma
   _{\rho a}^a-\frac{1}{2} g_{}^{\rho \nu } \Gamma _{\rho
   a}^a+g_{}^{a\nu } \Gamma _{a\rho }^{\rho
   }-g_{}^{a\rho } \Gamma _{a\rho }^{\mu }-g_{}^{\mu a}
   \Gamma _{\rho a}^{\rho }+g_{}^{\rho a} \Gamma _{\rho
   a}^{\nu }-g_{,\rho }^{\mu \rho }+g_{,\rho }^{\rho \nu }=0
$$
I cant see how this is equal to equation (2)
 A: The attempt of obtaining  
$$
g^{[\mu\nu]}_{,\nu}
    +\frac{1}{2}(
    {{\Gamma}}^{\rho}_{\rho\nu}-{{\Gamma}}^{\rho}_{\nu\rho}
    )
    g^{(\mu\nu)}
    =0,
$$
was almost right! The only thing missing was a little care with relabeling indices.
We will proceed in three main steps.
1.) So when contracting Eq. (1) with respect to $\mu$ and $\rho$, we get the identity:
$$
-\frac{1}{2} g^{\rho \nu } {\Gamma} _{a\rho}^{a}-\frac{1}{2} g^{\rho
   \nu} \Gamma _{\rho a}^{a}+g^{a\nu} \Gamma _{a\rho}^{\rho}+g^{\rho a} \Gamma _{\rho a}^{\nu}+g_{,\rho}^{\rho\nu}=0.
$$
Now by relabeling the dummy indices in the 3rd term as $ a \leftrightarrow \rho$, we get
that the 3rd term can be written as $g^{\rho \nu} \Gamma _{\rho a}^{a}$. Moreover, we 
can see that now the 2nd and 3rd term can be simplified: adding them together gives 
$+ \tfrac{1}{2}g^{\rho  \nu} \Gamma _{\rho a}^{a}$. By a final index label change $\nu \to \mu$, we get that:
$$
-\frac{1}{2} g^{\rho \mu } {\Gamma} _{a\rho}^{a}+\frac{1}{2} g^{\rho
   \mu} \Gamma _{\rho a}^{a}+g^{\rho a} \Gamma _{\rho a}^{\mu}+g_{,\rho}^{\rho\mu}=0. \; \; \; \; (A)
$$
2.) When contracting Eq. (1) with respect to $\nu$ and $\rho$, we get the identity:
$$
-\frac{1}{2} g^{\mu \rho } \Gamma _{a\rho}^a-\frac{1}{2} g^{\mu
   \rho} \Gamma _{\rho a}^a+g_{}^{a\rho } \Gamma _{a\rho}^{\mu }+g^{\mu a} \Gamma _{\rho a}^{\rho
   }+g_{,\rho }^{\mu \rho }=0.
$$
Let us also rename the dummy indices in the 4th term as $ a \leftrightarrow \rho$. We can see now that the 4th term is simply $g^{\mu \rho } \Gamma _{a\rho}^a$, and the 1st and 4th term hence together give $\frac{1}{2} g^{\mu \rho } \Gamma _{a\rho}^a$. Moreover, let us perform also the $ a \leftrightarrow \rho$ "dummy index relabeling", yielding $g_{}^{\rho a} \Gamma _{\rho a}^{\mu }$ for the 3rd term. After these manipulations our identity reads as
$$
\frac{1}{2} g^{\mu \rho } \Gamma _{a\rho}^a-\frac{1}{2} g^{\mu
   \rho} \Gamma _{\rho a}^a+g_{}^{\rho a} \Gamma _{\rho a}^{\mu }+g_{,\rho }^{\mu \rho }=0. \; \; \; \; (B)
$$
3.) Now taking (B)-(A), we obtain:
$$
g_{,\rho }^{\mu \rho } -g_{,\rho}^{\rho\mu} + \frac{1}{2}\left( g^{\mu \rho }  +  g^{\mu \rho }\right) \left( \Gamma _{a\rho}^a - \Gamma _{\rho a}^a \right)=0,
$$
which after the $a \to \rho$ and $\rho \to \nu$ relabeling is exactly the same as the desired Eq. (2).
