On genus of division algebras

The genus $gen(D)$ of a finite-dimensional central division algebra $D$ over a field $F$ is defined as the collection of classes $[D^{\mathrm{\prime }}]\in Br(F)$, where $D^{\mathrm{\prime }}$ is a central division $F$-algebra having the same maximal subfields as $D$. We show that the fact that quaternion division algebras $D$ and $D^{\mathrm{\prime }}$ have the same maximal subfields does not imply that the matrix algebras $M_l(D)$ and $M_l(D^{\mathrm{\prime }})$ have the same maximal subfields for $l>1$. Moreover, for any odd $n>1$, we construct a field $L$ such that there are two quaternion division $L$-algebras $D$ and $D^{\mathrm{\prime }}$ and a central division $L$-algebra $C$ of degree and exponent $n$ such that $gen(D)=gen(D^{\mathrm{\prime }})$ but $gen(D \otimes C) \ne gen(D’ \otimes C)$.