Linkage of quadratic Pfister forms

We study the necessary conditions for sets of quadratic n-fold Pfister forms to have a common (n-1)-fold Pfister factor. For any set S of n-fold Pfister forms generating a subgroup of (IqF)-F-n/Iqn+1F of order 2s in which every element has an n-fold Pfister representative, we associate an invariant in Iqn+1F which lives inside Iqn+s-1F when the forms in S have a common (n-1)-fold Pfister factor. We study the properties of this invariant and compute it explicitly in a few interesting cases.