Products of unipotent matrices of index 2 over division rings

Let D be a division ring. The first aim of this paper is to describe all unipotent matrices of index 2 in the general linear group \(\mathrm {GL}_n(D)\) ${\mathrm{GL}}_n(D)$ of degree n and in the Vershik–Kerov group \(\mathrm{GL} _{\rm VK}(D)\) ${\mathrm{GL}}_{\mathrm{VK}}(D)$ . As a corollary, the subgroups generated by such matrices are investigated. The next aim is to seek a positive integer d such that every matrix in these groups is a product of at most d unipotent matrices of index 2. For example, we show that if every element in the derived subgroup \(D'\) $D^{\prime }$ of \(D^*=D\backslash \{0\}\) $D^{\ast }=D\backslash \{0\}$ is a product of at most c commutators in \(D^*\) $D^{\ast }$ , then every matrix in \(\mathrm{GL}_n(D)\) ${\mathrm{GL}}_n(D)$ (resp., \(\mathrm{GL} _{\rm VK}(D)\) ${\mathrm{GL}}_{\mathrm{VK}}(D)$ , which is a product of some unipotent matrices of index 2, can be written as a product of at most 4+3c (resp.,5 + 3c) of unipotent matrices of index 2 in \(\mathrm{GL}_n(D)\) ${\mathrm{GL}}_n(D)$ (resp., \(\mathrm{GL}_{\rm VK}(D))\) ${\mathrm{GL}}_{\mathrm{VK}}(D))$ .