  In the decomposition of gauge-theory amplitudes into kinematic and color
factors, the color factors (at a given loop order $L$) span a proper subspace
of the extended trace space (which consists of single and multiple traces of
generators of the gauge group, graded by powers of $N$). Using an iterative
process, we systematically construct the $L$-loop color space of four-point
amplitudes of fields in the adjoint representation of SU($N$), SO($N$), or
Sp($N$). We define the null space as the orthogonal complement of the color
space. For SU($N$), we confirm the existence of four independent null vectors
(for $L \ge 2$) and for SO($N$) and Sp($N$), we establish the existence of
seventeen independent null vectors (for $L \ge 5$). Each null vector
corresponds to a group-theory constraint on the color-ordered amplitudes of the
gauge theory.
