Two-Term Polynomial Identities

We study algebras satisfying a two-term multilinear identity of the form x₁⋯ xₙ=qxσ(1)⋯ xσ(n). We show that such algebras with $q=1$ are eventually commutative in the sense that the equality x₁⋯ xₖ =xτ(1)⋯ xτ(k) holds for k large enough and all permutations τ∈ Sₖ. For various natural examples, we prove that k can be taken to be $n+1$ or $n+2$, but in other cases the minimal k can be as large as $2n-3$. In the case that q≠1, we establish that the algebra must be nilpotent.