Non-$R$-trivial proper projective similitudes in type $A_3\equiv D_3$

Abstract

Over an arbitrary field of characteristic different from $2$ admitting an anisotropic torsion $3$-fold Pfister form, we apply a construction due to Merkurjev to produce an algebra with orthogonal involution of degree $6$ which admits proper projective similitudes that are not $R$-trivial. In particular, such examples exist over every finitely generated transcendental extension of a local or global number field, as well as over every finitely generated extension of transcendence degree $3$ of $\mathbb{R}$.