Reduced $K$-theory for the group $ G(D)=D^*/\operatorname{Nrd}_D(D^*)D $

Let $D$ be a division algebra finite-dimensional over its center $F$. Put $D^*=D\setminus\{0\}$ and denote by $D $ the commutator subgroup of $D^*$. Let $G(D)=D^*/\operatorname{Nrd}_D(D^*)D $ where $\operatorname{Nrd}_D(D^*)$ is the image of $D^*$ under the reduced norm map. In this note we investigate the group $G(D)$ by two methods: by using valuation theory for division algebras and by using $K$-theory methods. This enables us to compute the group $G(D)$ in several cases. In particular, we give an explicit formula for $G(D)$ when $F$ is a local filed and $D$ is a tame division algebra