Quadratic forms of dimension 8 with trivial discrimiand and Clifford algebra of index 4

TL;DR

This paper provides a new proof for a classification result of certain 8-dimensional quadratic forms with specific algebraic properties, using a theorem on algebra decomposability with involution.

Contribution

It introduces a novel proof approach for classifying quadratic forms of dimension 8 with trivial discriminant and Clifford algebra of index 4, based on algebra decomposability theorems.

Findings

New proof of Izhboldin and Karpenko's classification theorem

Demonstrates decomposability of degree 8, index 4 algebras with orthogonal involution

Clarifies the structure of quadratic forms with specified algebraic invariants

Abstract

Izhboldin and Karpenko proved in 2000 that any quadratic form of dimension 8 with trivial discriminant and Clifford algebra of index 4 is isometric to the transfer, with respect to some quadratic \'etale extension, of a quadratic form similar to a 2-fold Pfister form. We give a new proof of this result, based on a theorem of decomposability for degree 8 and index 4 algebras with orthogonal involution.