The paper presents a kinematic identity satisfied by the kinematic factors of diagrams describing tree amplitudes in massless gauge theories. This identity is analogous to the Jacobi identity for color factors. Using this identity, new relations between color-ordered partial amplitudes are derived. The authors discuss applications to multiloop calculations via the unitarity method, particularly for two-loop four-point QCD amplitudes and gravity tree amplitudes. The identity is used to reorganize gravity tree amplitudes, offering new insights into the KLT relations between gauge and gravity tree amplitudes. This can be used to derive novel relations similar to the KLT ones, which may be helpful in higher-loop studies of the ultraviolet properties of gravity theories.The paper presents a kinematic identity satisfied by the kinematic factors of diagrams describing tree amplitudes in massless gauge theories. This identity is analogous to the Jacobi identity for color factors. Using this identity, new relations between color-ordered partial amplitudes are derived. The authors discuss applications to multiloop calculations via the unitarity method, particularly for two-loop four-point QCD amplitudes and gravity tree amplitudes. The identity is used to reorganize gravity tree amplitudes, offering new insights into the KLT relations between gauge and gravity tree amplitudes. This can be used to derive novel relations similar to the KLT ones, which may be helpful in higher-loop studies of the ultraviolet properties of gravity theories.