Pfister's Theorem for orthogonal involutions of degree 12

TL;DR

This paper generalizes Pfister's theorem for quadratic forms of dimension 12 within the third power of the fundamental ideal to orthogonal involutions, leveraging the properties of a half-spin representation of Spin(12).

Contribution

It introduces a new generalization of Pfister's theorem to orthogonal involutions of degree 12 using representation theory.

Findings

Established a link between half-spin representations and orthogonal involutions.

Extended Pfister's theorem to a broader class of algebraic structures.

Provided new insights into the structure of quadratic forms and involutions.

Abstract

We use the fact that a projective half-spin representation of $Spin_{12}$ has an open orbit to generalize Pfister's result on quadratic forms of dimension 12 in $I^3$ to orthogonal involutions.