Relatively free associative Lie nilpotent algebras of rank $3$

Let $\Phi$ be an arbitrary unital associative and commutative ring. The relatively free Lie nilpotent algebras with three generators over $\Phi$ are studied. The product theorem is proved: $T^{(n)}T^{(m)} \subseteq T^{(n + m-1)}$, where $T^{(n)}$ is a verbal ideal generated by the commutators of degree $n$. The identities of three variables that are satisfied in a free associative Lie nilpotent algebra of degree $n\geq 3$ are described. It is proved that the additive structure of the considered algebra is a free module over the ring $\Phi$.