Triality over fields with $I^3=0$

TL;DR

This work analyzes trialitarian triples of degree $8$ with quadratic pairs, providing an explicit isotropy criterion over fields where $I_q^3F=0$ and connecting isotropy to the Schur indices of the three involved algebras. It extends the framework of quadratic extensions of algebras with unitary involution to characteristic $2$ and demonstrates that Ad$(q_h)$ forms a quadratic extension of Ad$(h)$, with discriminant and Clifford-invariant behavior governed by the base unitary involution. The results yield a precise classification of isotropic triples (allowing only certain index patterns) and show how anisotropic triples can still occur in characteristic $2$ via explicit constructions, broadening understanding of the interaction between involutions, Clifford algebras, and cohomological restrictions. These findings have implications for the classification of related algebraic groups and their torsors over fields of mixed characteristic.

Abstract

We characterize isotropic trialitarian triples in terms of the Schur indices of the underlying algebras over a base field $F$ of arbitrary characteristic satisfying $I_q^3 F=0$. We also construct anisotropic trialitarian triples over such fields.

Theorems & Definitions (23)

• Proposition 2.1
• Definition 3.1
• Remark 3.2
• Example 3.3
• Example 3.4
• Proposition 3.5
• proof
• Proposition 3.6
• proof
• Lemma 3.7