Contorsion tensor and notation I was reading an article about Einstein-Cartan universes and the beginning he defines the contortion tensor as $$ K^a\vphantom{K}_{bc}=S^a\vphantom{S}_{bc}+S_{bc}\vphantom{S}^a+S_{cb}\vphantom{S}^a $$ with $$S^a\vphantom{S}_{bc}=\Gamma^a\vphantom{S}_{[bc]}.$$
Then he writes two equalities that I cannot understand:
$$K_{(a|b|c)}=-2S_{(ac)b}$$ and $${    }K_{[a|b|c]}=-S_{bac}. $$

What are the definition of  $K_{(a|b|c)}$ and ${    }K_{[a|b|c]}$? 

 A: Obviously, $K_{abc} = g_{ad}{K^d}_{bc}$.  And I'm guessing you know that $()$ means symmetrization and $[]$ means antisymmetrization over whatever indices are contained between them.
The vertical bars are there to "suspend" the symmetrization or anti-symmetrization that is happening on the other indices.  That is, the $b$ in each case is not involved:
\begin{gather}
  K_{(a|b|c)} = \frac{1}{2} \left( K_{abc} + K_{cba} \right),
  \\
  K_{[a|b|c]} = \frac{1}{2} \left( K_{abc} - K_{cba} \right).
\end{gather}
So, for example, it's straightforward to compute
\begin{align}
  K_{(a|b|c)} &= \frac{1}{2} \left( K_{abc} + K_{cba} \right)
  \\
  &= \frac{1}{2} \left( S_{abc} + S_{bca} + S_{cba} + S_{cba} + S_{bac} + S_{abc} \right)
  \\
  &= S_{abc} + S_{b(ca)} + S_{cba}
  \\
  &= S_{abc} + S_{cba}
  \\
  &= -(S_{acb} + S_{cab})
  \\
  &= -2S_{(ac)b}
\end{align}
In the third line, I eliminated $S_{b(ca)}$ because it is totally antisymmetric on the last two indices (by definition), so the symmetric part of those two indices is zero.  In the fourth line, I swapped the last two indices, and flipped the sign, because of that same antisymmetry.
Similarly, we have
\begin{align}
  K_{[a|b|c]} &= \frac{1}{2} \left( K_{abc} - K_{cba} \right)
  \\
  &= \frac{1}{2} \left( S_{abc} + S_{bca} + S_{cba} - S_{cba} - S_{bac} - S_{abc} \right)
  \\
  &= \frac{1}{2} \left( S_{bca} - S_{bac} \right)
  \\
  &= S_{b[ca]}
  \\
  &= -S_{b[ac]}
  \\
  &= -S_{bac}
\end{align}
All of these manipulations should become very obvious once you get a little practice with index gymnastics.
