  The duality between color and kinematics enables the construction of
multiloop gravity integrands directly from corresponding gauge-theory
integrands. This has led to new nontrivial insights into the structure of
gravity theories, including the discovery of enhanced ultraviolet
cancellations. To continue to gain deeper understandings and probe these new
properties, it is crucial to further improve techniques for constructing
multiloop gravity integrands. In this paper, we show by example how one can
alleviate difficulties encountered at the multiloop level by relaxing the
color-kinematics duality conditions to hold manifestly only on unitarity cuts
instead of globally on loop integrands. As an example, we use a minimal ansatz
to construct an integrand for the two-loop four-point nonsupersymmetric pure
Yang-Mills amplitude in $D$ dimensions that is compatible with these relaxed
color-kinematics duality constraints. We then immediately obtain a
corresponding gravity integrand through the double-copy procedure. Comments on
ultraviolet divergences are also included.
