Outer forms of type $A_2$ with infinite genus

Abstract

Let $G$ be an absolutely almost simple algebraic group over a field $K$. The genus ${\mathbf{gen}}_K(G)$ of $G$ is the set of $K$-isomorphism classes of $K$-forms $G^{\prime }$ of $G$ that have the same $K$-isomorphism classes of maximal $K$-tori as $G$. We construct an example of outer forms of type $A_2$ with infinite genus.