Let $F$ be the free algbra generated by letters $a$ and $b$, endow it with its usual grading, and let $I$ be the ideal of $F$ generated by all elements of the form $[[x,y],z]$ with $x$, $y$ and $z$ in $F$. Since the iterated bracket is trilinear, it is enough to consider homogeneous elements $x$, $y$ and $z$ when generating $I$, and therefore $I$ is a homogeneous ideal. The ring $F/I$ satisfies the desired identity.
We want to show that $R$ is non trivial and non-commutative. If the three letters are homogeneous, the element $[[x,y],z]$ cannot be nonzero and have degree $\leq 2$. Indeed, that can only happen if one of $x$, $y$ or $z$ has degree zero, and then the iterated bracket is itself zero. It follows that $I$ does not contain $[a,b]$, so that $R$ is nontrivial and noncommutative.