Products of Commutators of Involutions in Skew Linear Groups

In connection with [Theorem 4.6, Linear Algebra Appl. 646, 119–131, (2022)], we show that each matrix in the commutator subgroup of the general linear group over a centrally-finite division ring D, in which each element in the commutator subgroup of D is a product of at most s commutators, can be written as a product of at most \(3+3\left\lceil \frac{s}{\lfloor n/2 \rfloor } \right\rceil \) $3+3⌈\frac{s}{\lfloor n/2\rfloor }⌉$ commutators of involutions if \(\mathrm {char\,}D\ne 2\) $\mathrm{char}D\ne 2$ , where \({\displaystyle \lceil x \rceil }\) ${\displaystyle \lceil x\rceil }$ , \({\displaystyle \lfloor x \rfloor }\) ${\displaystyle \lfloor x\rfloor }$ denote the ceiling and floor functions of x, respectively. Moreover, we also present the special case when \(D= \mathbb {H}\) $D=\mathbb{H}$ , the division ring of quaternions, and an application in real group algebras.