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