Similar to a previous question here, I wonder if cyclic permutations are the only relations amongst traces of (non-commutative) monomials. Since the evaluations $\operatorname{tr}:k\langle x,y,\dots \rangle \to k$ take an infinite dimensional vector space to a one-dimensional vector space there must be quite a few relations, but I wonder if any of them are on binomials other than the cyclic permutations.
At any rate, for small dimensions, we probably get some extra relations.
It appears that $\operatorname{tr}(AABABB−AABBAB) = 0$ for all $2×2$ matrices. Is this true? How does one prove it?