  The conjectured duality between color and kinematics has significantly
advanced our understanding of both gauge and gravitational theories. However,
constructing numerators that manifest the color-kinematics (CK) duality, even
for the two-loop four-gluon amplitude in pure Yang-Mills, has been challenging.
In this paper, we revisit this amplitude and show that the difficulty of
applying CK duality can be overcome by introducing a simple deformation. Our
approach distinguishes itself from previous studies by maximizing the use of
off-shell CK duality while maintaining a compact ansatz. In particular, the
deformation we introduce satisfies a subset of off-shell dual Jacobi relations.
The resulting numerators are presented in $d$-dimensionally Lorentz invariant
local form and are applicable to all helicities of external gluons. The
solution we provide can be directly employed to construct the corresponding
gravitational amplitude through double copy. Our findings suggest a novel and
efficient strategy for constructing high-loop gauge and gravitational
amplitudes using CK duality.
