  Tree-level amplitudes of gauge theories are expressed in a basis of auxiliary
amplitudes with only cubic vertices. The vertices in this formalism are
explicitly factorized in color and kinematics, clarifying the color-kinematics
duality in gauge theory amplitudes. The basis is constructed making use of the
KK and BCJ relations, thereby showing precisely how these relations underlie
the color-kinematics duality. We express gravity amplitudes in terms of a
related basis of color-dressed gauge theory amplitudes, with basis coefficients
which are permutation symmetric.
