Normal Quaternionic Matrices and Finitely Generated Witt Rings

We present a new approach to verify the Elementary Type Conjecture for abstract Witt rings with small number of square classes. To do so, we make use of an abstract analogue of the 2-torsion part of the Brauer group. This enables us to describe the entire structure of an abstract Witt ring with $2^n$ square classes in terms of a unique $n\times n$ matrix. Via computational search, we find all these matrices for $n$ up to $7$ with the exception of a few cases of $n=7$ associated with Witt rings of level at least $4$. All obtained results affirm the Elementary Type Conjecture.