  We explore tree-level amplitude relations for SU(N)xSU(M) bi-fundamental
matter theories. Embedding the group-theory structure in a Lie three-algebra,
we derive Kleiss-Kuijf-like relations for bi-fundamental matter theories in
general dimension. We investigate the three-algebra color-kinematics duality
for these theories. Unlike the Yang-Mills two-algebra case, the three-algebra
Bern-Carrasco-Johansson relations depend on the spacetime dimension and on the
detailed symmetry properties of the structure constants. We find the presence
of such relations in three and two dimensions, and absence in D>3.
Surprisingly, beyond six point, such relations are absent in the
Aharony-Bergman-Jafferis-Maldacena theory for general gauge group, while the
Bagger-Lambert-Gustavsson theory, and its supersymmetry truncations, obey the
color-kinematics duality like clockwork. At four and six points the relevant
partial amplitudes of the two theories are bijectively related, explaining
previous results in the literature. In D=2 the color-kinematics duality gives
results consistent with integrability of two-dimensional $\mathcal{N}=16$
supergravity: The four-point amplitude satisfies a Yang-Baxter equation; the
six- and eight-point amplitudes vanish for certain kinematics away from
factorization channels, as expected from integrability.
