  In this paper, we study off-shell currents built from the Jacobi identity of
the kinematic numerators of $gg\to X$ with $X=ss,q\bar{q},gg$. We find that
these currents can be schematically written in terms of three-point interaction
Feynman rules. This representation allows for a straightforward understanding
of the Colour-Kinematics duality as well as for the construction of the
building blocks for the generation of higher-multiplicity tree-level and
multi-loop numerators. We also provide one-loop integral relations through the
Loop-Tree duality formalism with potential applications and advantages for the
computation of relevant physical processes at the Large Hadron Collider. We
illustrate these integral relations with the explicit examples of QCD one-loop
numerators of $gg\to ss$.
