How much commutativity is needed to prove polynomial identities?

Let f be a non-commutative polynomial such that f = 0 if we assume that the variables in f commute. Let Q(f) be the smallest k such that there exist polynomials g1, g ′ 1, g2, g ′ 2, . . . , gk, g ′ k with f ∈ I([g1, g 1], [g2, g 2], . . . , [gk, g k]) , where [g, h] = gh − hg. Then Q(f) ≤ ` n 2 , where n is the number of variables of f . We show that there exists a polynomial f with Q(f) = Ω(n). We pose the problem of constructing such an f explicitly, pointing out that the solution may have applications to complexity of proofs.