Writing the fully color dressed and graviton amplitudes, respectively, as \documentclass[12pt]{minimal}
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\begin{document}$ \textbf{A} = \left\langle {C} \mathrel{\left | {\vphantom {C A}} \right. } {A} \right\rangle = \left\langle {C} \mathrel{\left | {\vphantom {C {M\left| N \right.}}} \right. } {{M\left| N \right.}} \right\rangle $\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$ {\textbf{A}_{gr}} = \left\langle {{\tilde{N}}} \mathrel{\left | {\vphantom {{\tilde{N}} {M\left| N \right.}}} \right. } {{M\left| N \right.}} \right\rangle $\end{document}, where \documentclass[12pt]{minimal}
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\begin{document}$ \left| A \right\rangle $\end{document} is a set of Kleiss-Kuijf color ordered basis, \documentclass[12pt]{minimal}
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\begin{document}$ \left| N \right\rangle $\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$ \left| {\tilde{N}} \right\rangle $\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$ \left| C \right\rangle $\end{document} are the similarly ordered numerators and color coefficients, we show that the propagator matrix M has (n − 3)(n − 3)! independent eigenvectors \documentclass[12pt]{minimal}
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\begin{document}$ \left| {\lambda_j^0} \right\rangle $\end{document} with zero eigenvalue, for n-particle processes. The resulting equations \documentclass[12pt]{minimal}
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\begin{document}$ \left| {\lambda_j^0} \right\rangle = 0 $\end{document} are relations among the color ordered amplitudes. The freedom to shift \documentclass[12pt]{minimal}
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\begin{document}$ \left| N \right\rangle \to \left| N \right\rangle + \sum\nolimits_j {{f_j}\left| {\lambda_j^0} \right\rangle } $\end{document} and similarly for \documentclass[12pt]{minimal}
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\begin{document}$ \left| {\tilde{N}} \right\rangle $\end{document} where fj are (n − 3)(n − 3)! arbitrary functions, encodes generalized gauge transformations. They yield both BCJ amplitude and KLT relations, when such freedom is accounted for. Furthermore, fj can be promoted to the role of effective Lagrangian vertices in the field operator space.
