  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 $\operatorname{char}(F) \neq 2$
and $u(F)=4$, then every three quaternion algebras share a common maximal
subfield.
