A note on commutators in the group of infinite triangular matrices over a ring

We investigate the commutators of elements of the group (Formula presented.) of infinite unitriangular matrices over an associative ring (Formula presented.) with (Formula presented.) and a commutative group (Formula presented.) of invertible elements. We prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. As a direct consequence, we give a complete characterization of the lower central series of the group (Formula presented.) including the width of its terms with respect to basic commutators and Engel words. With an additional restriction on the ring (Formula presented.) , we show that the derived subgroup of (Formula presented.) coincides with the group (Formula presented.). These results generalize the results obtained for triangular groups over a field.