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Tree-level amplitudes in N=4 SYM can be decomposed into partial orcolor-ordered amplitudes. Identities relating various partial amplitudes havebeen known since the 80's. They are Kleiss-Kuijf (KK) identities. In 2008,Bern, Carrasco and Johansson (BCJ) introduced a new set of identities whichreduce the number of independent partial amplitudes to (n-3)!. In recent years,several formulations for partial amplitudes have been discovered and shown tobe equivalent to each other. These can be thought of as simple dualities in thesense that different formulations make manifest different properties of thesame object; the amplitude. One such formulation is the ACCK Grassmannianformulation which makes Yangian invariance of individual partial amplitudesmanifest. A different formulation is the so-called connected formula introducedby Witten in twistor space and formulated in momentum space by Roiban, Spradlinand Volovich. It has been argued that the connected formula is ideal forstudying properties which are related to the full amplitude, such as the KKrelations, and not to particular partial amplitudes, like Yangian invariance.Following this logic, it is very natural to expect that the BCJ identitiesshould have a very simple proof in the connected formulation. In this shortnote we show that this is indeed the case.

Fundamental BCJ Relation in N=4 SYM From The Connected Formulation