Instances of the Kaplansky–Lvov multilinear conjecture for polynomials of degree three

Given a positive integer d  , the Kaplansky–Lvov conjecture states that the set of values of a multilinear noncommutative polynomial f∈C〈x1,…,xn〉f∈C〈x1,…,xn〉 on the matrix algebra Md(C)Md(C) is a vector subspace. In this article the technique of using one-wiggle families of Sylvester's clock-and-shift matrices is championed to establish the conjecture for polynomials f of degree three when d   is even or d<17d<17.