We analyze theories with color-kinematics duality from an algebraic perspective and find that any such theory has an underlying BV${}^{\color{gray} \blacksquare}$-algebra structure, extending the ideas of arXiv:1912.03110. Conversely, we show that any theory with a BV${}^{\color{gray} \blacksquare}$-algebra features a kinematic Lie algebra that controls interaction vertices, both on- and off-shell. We explain that the archetypal example of a theory with BV${}^{\color{gray} \blacksquare}$-algebra is Chern-Simons theory, for which the resulting kinematic Lie algebra is isomorphic to the Schouten-Nijenhuis algebra on multivector fields. The BV${}^{\color{gray} \blacksquare}$-algebra implies the known color-kinematics duality of Chern-Simons theory. Similarly, we show that holomorphic and Cauchy-Riemann (CR) Chern-Simons theories come with BV${}^{\color{gray} \blacksquare}$-algebras and that, on the appropriate twistor spaces, these theories organize and identify kinematic Lie algebras for self-dual and full Yang-Mills theories, as well as the currents of any field theory with a twistorial description. We show that this result extends to the loop level under certain assumptions.

v2: presentation improved, typos fixed, published version