Products of nonprimary cyclic conjugacy classes in the general linear group

A cyclic square matrix (and its conjugacy class) C over a field K is called (m,k)-cyclic if it has a decomposition $C=A\oplus B$, where $\dim A=m,\dim B=k$ and $m,k\mathrm{\ne }0$. It is shown that the product of two nonsingular (m,k)-cyclic conjugacy classes $\mathrm{\Omega }$ and $\mathrm{\Psi }$ of GL(m+k,K) contains all nonscalar matrices P in GL(m+k,K) with determinant $\det P=\det \mathrm{\Omega }\mathrm{\Psi }$.