On the images of multilinear maps of matrices over finite-dimensional division algebras

Let R be a central simple algebra finite-dimensional over its center F of characteristic 0. We will show that every element of reduced trace 0 in R can be expressed as [a, [c, b]] + lambda[c, [a, b]] for some a, b, c is an element of R where lambda not equal 0, -1. In addition, let D be a division algebra satisfying the conditions above. We will also show that the set of values of any nonzero multilinear polynomial of degree at most three, with coefficients from the center F of D, evaluated on M-k(D), k >= 2, contains all matrices of reduced trace 0. (C) 2015 Elsevier Inc. All rights reserved.