Symmetric split for third rank tensor? I know that for arbitrary second rank tensor we have
$$
A_{\mu\nu}=A_{(\mu\nu)}+A_{[\mu\nu]}\quad ,
$$
where $A_{(\mu\nu)}=\frac 1 2(A_{\mu\nu}+ A_{\nu\mu}),\quad A_{[\mu\nu]}=\frac 1 2(A_{\mu\nu}-A_{\nu\mu}) $.
The question is: How about third rank tensors?  Can we have a formula like $A_{\alpha\beta\gamma}=A_{\alpha(\beta\gamma)}+....-....\pm A_{[\alpha\beta]\gamma}$ for example?
 A: \begin{eqnarray}
 T_{[{\mu\nu}\alpha]}&=& T_{{\rho\sigma}\beta}\delta^\rho_{[\mu}\delta^\sigma_\nu\delta^\beta_{\alpha]}\;,\\
    &=& \frac 1 6 T_{{\rho\sigma}\beta} \bigg[   \delta^\rho_{\mu}\delta^\sigma_\nu\delta^\beta_{\alpha}- \delta^\rho_{\mu}\delta^\sigma_\alpha\delta^\beta_{\nu}+\delta^\rho_{\alpha}\delta^\sigma_\mu\delta^\beta_{\nu}  -\delta^\rho_{\alpha}\delta^\sigma_\nu\delta^\beta_{\mu}   +\delta^\rho_{\nu}\delta^\sigma_\alpha\delta^\beta_{\mu} -\delta^\rho_{\nu}\delta^\sigma_\mu\delta^\beta_{\alpha}    \bigg]\;,\\
    &=& \begin{cases}
         &\frac 1 3 \big\{ T_{\mu[\nu\alpha]} +T_{\nu[\alpha\mu]}  +T_{\alpha[\mu\nu]}    \big\} \\
         &\frac 1 3 \big\{ T_{[\mu\nu]\alpha} +T_{[\nu\alpha]\mu}  +T_{[\alpha\mu]\nu}    \big\} \\
         &\frac 1 3 \big\{ T_{[\mu|\nu|\alpha]} +T_{[\nu|\alpha|\mu]}  +T_{[\alpha|\mu|\nu]}    \big\} 
        \end{cases}\;,\quad (3)\\
        &=& \frac 1 6 \bigg[ T_{\mu\nu\alpha}-T_{\mu\alpha\nu} +T_{\alpha\mu\nu}-T_{\alpha\nu\mu} +T_{\nu\alpha\mu}-T_{\nu\mu\alpha}  \bigg]\quad (1)
\end{eqnarray}
\begin{eqnarray}
 T_{({\mu\nu}\alpha)}&=& T_{{\rho\sigma}\beta}\delta^\rho_{(\mu}\delta^\sigma_\nu\delta^\beta_{\alpha)}\;,\\
    &=& \frac 1 6 T_{{\rho\sigma}\beta} \bigg[   \delta^\rho_{\mu}\delta^\sigma_\nu\delta^\beta_{\alpha}+ \delta^\rho_{\mu}\delta^\sigma_\alpha\delta^\beta_{\nu}+\delta^\rho_{\alpha}\delta^\sigma_\mu\delta^\beta_{\nu}  +\delta^\rho_{\alpha}\delta^\sigma_\nu\delta^\beta_{\mu}   +\delta^\rho_{\nu}\delta^\sigma_\alpha\delta^\beta_{\mu} +\delta^\rho_{\nu}\delta^\sigma_\mu\delta^\beta_{\alpha}    \bigg]\;,\\
    &=& \begin{cases}
         &\frac 1 3 \big\{ T_{\mu(\nu\alpha)} +T_{\nu(\alpha\mu)}  +T_{\alpha(\mu\nu)}    \big\} \\
         &\frac 1 3 \big\{ T_{(\mu\nu)\alpha} +T_{(\nu\alpha)\mu}  +T_{(\alpha\mu)\nu}    \big\} \\
         &\frac 1 3 \big\{ T_{(\mu|\nu|\alpha)} +T_{(\nu|\alpha|\mu)}  +T_{(\alpha|\mu|\nu)}    \big\} 
        \end{cases}\;,\quad (4)\\
        &=& \frac 1 6 \bigg[ T_{\mu\nu\alpha}+T_{\mu\alpha\nu} +T_{\alpha\mu\nu}+T_{\alpha\nu\mu} +T_{\nu\alpha\mu}+T_{\nu\mu\alpha}  \bigg]\quad (2)
\end{eqnarray}
From (1)(2) we have
\begin{equation}
 T_{[{\mu\nu}\alpha]}+T_{({\mu\nu}\alpha)} = \frac 1 3 \big[ T_{\mu\nu\alpha} +T_{\alpha\mu\nu} +T_{\nu\alpha\mu}  \big]
\end{equation}
Further, we using the first line of (3) and the second line of (4) for this relation we have
\begin{equation}
 T_{\mu[\nu\alpha]} +T_{\nu[\alpha\mu]}  +T_{\alpha[\mu\nu]} + T_{(\mu\nu)\alpha} +T_{(\nu\alpha)\mu}  +T_{(\alpha\mu)\nu} = T_{\mu\nu\alpha} +T_{\alpha\mu\nu} +T_{\nu\alpha\mu}\;,
\end{equation}
implies
\begin{equation}\boxed{
 T_{\mu\nu\alpha} = T_{\mu[\nu\alpha]} +T_{\nu[\alpha\mu]}  +T_{\alpha[\mu\nu]} + T_{(\mu\nu)\alpha} +T_{(\nu\alpha)\mu}  +T_{(\alpha\mu)\nu} -T_{\alpha\mu\nu} -T_{\nu\alpha\mu}}\;.
\end{equation}
Indeed, there are another eight ways to write this. But, start from this we can write
\begin{equation}\boxed{
 T_{\mu\nu\alpha} = T_{\mu[\nu\alpha]} +T_{\nu[\alpha\mu]}  -T_{\alpha[\mu\nu]} + T_{(\mu\nu)\alpha} -T_{(\nu\alpha)\mu}  +T_{(\alpha\mu)\nu}} \;.\label{ssp5}
\end{equation}
The another equivalent form of symmetric splits are
\begin{equation}
 T_{\mu\nu\alpha} \begin{cases}
                 & = T_{\mu[\nu\alpha]} -T_{\nu[\alpha\mu]}  +T_{\alpha[\mu\nu]} + T_{(\mu|\nu|\alpha)} +T_{(\nu|\alpha|\mu)}  -T_{(\alpha|\mu|\nu)}\\
                 & = T_{[\mu\nu]\alpha} -T_{[\nu\alpha]\mu}  +T_{[\alpha\mu]\nu} + T_{\mu(\nu\alpha)} +T_{\nu(\alpha\mu)}  -T_{\alpha(\mu\nu)}\\
                 & = T_{[\mu\nu]\alpha} +T_{[\nu\alpha]\mu}  -T_{[\alpha\mu]\nu} + T_{(\mu|\nu|\alpha)} -T_{(\nu|\alpha|\mu)}  +T_{(\alpha|\mu|\nu)}\\
                 & =  T_{[\mu|\nu|\alpha]} +T_{[\nu|\alpha|\mu]}  -T_{[\alpha|\mu|\nu]} + T_{\mu(\nu\alpha)} -T_{\nu(\alpha\mu)}  +T_{\alpha(\mu\nu)}\\
                 & =  T_{[\mu|\nu|\alpha]} -T_{[\nu|\alpha|\mu]}  +T_{[\alpha|\mu|\nu]} + T_{(\mu\nu)\alpha} +T_{(\nu\alpha)\mu}  -T_{(\alpha\mu)\nu}
                \end{cases}\label{fivet}
\end{equation}
