Schouten[exp] attempts to automatically remove spurious terms in exp by applying the Schouten’s identity.
Schouten applies the identity for 4-vectors on at most 42 terms in a sum. If it should operate on a larger expression you can give a second argument, e.g. Schouten[expr, 4711] which will work on sums with less than 4711 terms.
Schouten is also an option of Contract and DiracTrace. It may be set to an integer indicating the maximum number of terms onto which the function Schouten will be applied.
Overview, Contract, DiracTrace, FCSchoutenBruteForce.
((LC[\[Mu], \[Nu], \[Rho], \[Sigma]] FV[p, \[Tau]] + LC[\[Nu], \[Rho], \[Sigma], \[Tau]] FV[p, \[Mu]] + LC[\[Rho], \[Sigma], \[Tau], \[Mu]] FV[p, \[Nu]] +
LC[\[Sigma], \[Tau], \[Mu], \[Nu]] FV[p, \[Rho]] + LC[\[Tau], \[Mu], \[Nu], \[Rho]] FV[p, \[Sigma]]))
Schouten[%]\overline{p}^{\tau } \bar{\epsilon }^{\mu \nu \rho \sigma }+\overline{p}^{\mu } \bar{\epsilon }^{\nu \rho \sigma \tau }+\overline{p}^{\nu } \bar{\epsilon }^{\rho \sigma \tau \mu }+\overline{p}^{\rho } \bar{\epsilon }^{\sigma \tau \mu \nu }+\overline{p}^{\sigma } \bar{\epsilon }^{\tau \mu \nu \rho }
0