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 $I_q^{n}F\mathrm{/}{I}_q^{n+1}F$ of order $2^s$ in which every element has an $n$-fold Pfister representative, we associate an invariant in $I_q^{n+1}F$ which lives inside $I_q^{n+s-1}F$ 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.