An improved term ordering for the ideal of commuting matrices

Let X and Y be generic n by n matrices of indeterminatesx1,…,xr, y1,…,yr. Let S=k[x1,…,xr,y1,…,yr] where k is a field and r=n2. Let I S be the ideal generated by the entries of the matrix XY−YX. We will show how an old result on a good term order for this ideal can be improved using elementary methods.