Octonion algebras over rings are not determined by their norms

TL;DR

This paper provides examples of octonion algebras over rings that have isometric norms but are not isomorphic, answering a question about their classification.

Contribution

It demonstrates that octonion algebras over rings are not uniquely determined by their norms, offering new insights into their structure.

Findings

Octonion algebras with isometric norms can be non-isomorphic.

Provides explicit examples over rings.

Addresses an open question by H. Petersson.

Abstract

Answering a question of H. Petersson, we provide a class of examples of pair of octonion algebras over a ring having isometric norms.