Commutators of trace zero matrices over principal ideal rings

We prove that for every trace zero square matrix A of size at least 3 over a principal ideal ring R, there exist trace zero matrices X, Y over R such that XY ? YX = A. Moreover, we show that X can be taken to be regular mod every maximal ideal of R. This strengthens our earlier result that A is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is simpler than the earlier one.