This paper presents a kinematic identity satisfied by the kinematic factors of diagrams describing the tree amplitudes of massless gauge theories. This identity is a kinematic analog of the Jacobi identity for color factors. Using this identity, the authors find new relations between color-ordered partial amplitudes. They discuss applications to multiloop calculations via the unitarity method, particularly illustrating the relations between different contributions to a two-loop four-point QCD amplitude. The identity is also used to reorganize gravity tree amplitudes diagram by diagram, offering new insight into the structure of the KLT relations between gauge and gravity tree amplitudes. This can be used to obtain novel relations similar to the KLT ones. The paper also discusses the implications of this identity for higher-loop studies of the ultraviolet properties of gravity theories. The paper begins with an introduction to gauge and gravity scattering amplitudes, highlighting their simpler and richer structure compared to Feynman rules or Lagrangians. It discusses the Parke-Taylor maximally helicity violating (MHV) amplitudes in QCD, the delta-function support of amplitudes on polynomial curves in twistor space, and the Kawai-Lewellen-Tye (KLT) relations between gravity and gauge-theory tree amplitudes. The paper then discusses the color-ordered partial amplitudes and their simplifying relations dictated by the color algebra. It introduces the photon decoupling identity and the Kleiss-Kuijf relations, which reduce the number of independent n-point tree partial amplitudes. The paper presents a kinematic identity that further constrains the n-point color-ordered partial amplitudes at tree-level. This identity is based on the observation that gauge-theory amplitudes can be rearranged into a form where the kinematic factors of diagrams describing the amplitudes satisfy an identity analogous to the Jacobi identity obeyed by the color factors associated with the same diagrams. The paper discusses the implications of this identity for higher-point tree amplitudes and its applications to multiloop calculations via the unitarity method. It also discusses the implications of this identity for gravity tree amplitudes, showing how it clarifies the KLT relations and can be used to derive new representations for gravity tree amplitudes in terms of gauge-theory ones. The paper concludes with a discussion of the implications of the kinematic identity for higher-point tree amplitudes and its applications to multiloop calculations. It also discusses the implications of the identity for the structure of multiloop scattering amplitudes and the ultraviolet properties of gravity theories. The paper emphasizes the importance of the kinematic identity in understanding the structure of gauge and gravity amplitudes and its potential applications in higher-loop studies.This paper presents a kinematic identity satisfied by the kinematic factors of diagrams describing the tree amplitudes of massless gauge theories. This identity is a kinematic analog of the Jacobi identity for color factors. Using this identity, the authors find new relations between color-ordered partial amplitudes. They discuss applications to multiloop calculations via the unitarity method, particularly illustrating the relations between different contributions to a two-loop four-point QCD amplitude. The identity is also used to reorganize gravity tree amplitudes diagram by diagram, offering new insight into the structure of the KLT relations between gauge and gravity tree amplitudes. This can be used to obtain novel relations similar to the KLT ones. The paper also discusses the implications of this identity for higher-loop studies of the ultraviolet properties of gravity theories. The paper begins with an introduction to gauge and gravity scattering amplitudes, highlighting their simpler and richer structure compared to Feynman rules or Lagrangians. It discusses the Parke-Taylor maximally helicity violating (MHV) amplitudes in QCD, the delta-function support of amplitudes on polynomial curves in twistor space, and the Kawai-Lewellen-Tye (KLT) relations between gravity and gauge-theory tree amplitudes. The paper then discusses the color-ordered partial amplitudes and their simplifying relations dictated by the color algebra. It introduces the photon decoupling identity and the Kleiss-Kuijf relations, which reduce the number of independent n-point tree partial amplitudes. The paper presents a kinematic identity that further constrains the n-point color-ordered partial amplitudes at tree-level. This identity is based on the observation that gauge-theory amplitudes can be rearranged into a form where the kinematic factors of diagrams describing the amplitudes satisfy an identity analogous to the Jacobi identity obeyed by the color factors associated with the same diagrams. The paper discusses the implications of this identity for higher-point tree amplitudes and its applications to multiloop calculations via the unitarity method. It also discusses the implications of this identity for gravity tree amplitudes, showing how it clarifies the KLT relations and can be used to derive new representations for gravity tree amplitudes in terms of gauge-theory ones. The paper concludes with a discussion of the implications of the kinematic identity for higher-point tree amplitudes and its applications to multiloop calculations. It also discusses the implications of the identity for the structure of multiloop scattering amplitudes and the ultraviolet properties of gravity theories. The paper emphasizes the importance of the kinematic identity in understanding the structure of gauge and gravity amplitudes and its potential applications in higher-loop studies.