Triple Linkage of Quadratic Pfister Forms

Given a field $F$ of characteristic 2, we prove that if every three quadratic $n$-fold Pfister forms have a common quadratic $(n-1)$-fold Pfister factor then $I_q^{n+1}F=0$. As a result, we obtain that if every three quaternion algebras over $F$ share a common maximal subfield then $u(F)$ is either $0,2$ or $4$. We also prove that if $F$ is a nonreal field with $\mathrm{char}(F)\mathrm{\ne }2$ and $u(F)=4$, then every three quaternion algebras share a common maximal subfield.