Note on the non-adjacent BCFW deformations

Abstract: As the first part, we provide a proof of the consistency condition by Kleiss-Kuijf relation and the bonus relation by BCJ relation for the non-adjacent BCFW deformations. On the other hand, rather than appealing to field theory argument, we provide an alternative proof of bonus relation for the non-adjacent BCFW deformations by a purely S matrix analysis in the context of $\mathcal{N}=4$ SYM theory.