Products of commutators on a general linear group over a division algebra

We consider the word maps $\widetilde w\colon\mathrm{GL}_m(D)^{2k}\to\mathrm{GL}_n(D)$ and $\widetilde w\colon D^{*2k}\to D^*$ for a word $w=\prod_{i=1}^k[x_i,y_i]$, where $D$ is the division algebra over a field $K$. If $\widetilde w(D^{*2k})=[D^*,D^*]$ we prove that $\widetilde w(\mathrm{GL}_n(D))\supset E_n(D)\setminus Z(E_n(D))$, where $E_n(D)$ is the subgroup of $\mathrm{GL}_n(D)$ which is generated by transvections and $Z(E_n(D))$ is its center. If, in addition, $n>2$, we prove $\widetilde w(E_n(D))\supset E_n(D)\setminus Z(E_n(D))$.
The proof of the result is based on an analogue of the “Gauss decomposition with prescribed semisimple part” (see, J. Algebra 229 (2000), no. 1, 314–332) of the group $\mathrm{GL}_n(D)$ which is also is considered in this paper.