Let D be a division algebra and let \(n\) be a positive integer greater than 1. Assume that the commutator width \(\omega (D)\) of \(D^{*}\) is positive and finite. First, we revisit a question posed by F. S. Cater concerning the decomposition of matrices into a product of reflections over a division ring. Among results, we show that every matrix \(A\) in the special linear group \(\textrm{SL}_n(D)\) can be expressed as a product of at most \(\textrm{rank} (A-\mathrm I_n)+4\omega (D)\) reflections. Next, we study the decomposition of matrices in \(\textrm{SL}_n(D)\) into products of commutators of reflections in the general linear group \(\textrm{GL}_n(D)\) .
