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

TL;DR

This paper constructs examples of algebraic structures with orthogonal involutions that have proper projective similitudes not equivalent to the identity, over various fields including number fields and real extensions.

Contribution

It introduces a method to produce algebras with orthogonal involutions exhibiting non-$R$-trivial proper projective similitudes in type A3, expanding known examples across different fields.

Findings

Existence of such algebras over every finitely generated transcendental extension of local or global number fields.

Existence over every finitely generated extension of transcendence degree 3 of .

Construction based on Merkurjev's method applied to anisotropic torsion 3-fold Pfister forms.

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 $\mathbb{R}$.