Vector product algebras

TL;DR

This paper discusses the unique dimensions where vector product operations exist and provides a simple proof of their classification based on symmetric bilinear forms.

Contribution

It offers a concise, elementary proof of the classical result classifying vector product algebras in specific dimensions.

Findings

Vector products exist only in dimensions 0, 1, 3, and 7.

Isomorphism types are determined by symmetric bilinear forms.

Provides a simplified proof of a classical mathematical theorem.

Abstract

Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result.