A completion of our earlier work on the Cauchy problem for non-effectively hyperbolic operators

For hyperbolic differential operators $P$ with non-effectively hyperbolic double characteristics, we study the relationship between the Gevrey well-posedness threshold for strong well-posedness and the associated Hamilton map and flow. In our previous work, we showed that if the Hamilton map has a Jordan block of size $4$ on the double characteristic manifold $\mathrm{\Sigma }$ of codimension $3$, then the Cauchy problem for $P$ is well-posed in the Gevrey class $1<s<3$ for all lower-order terms, and that this result is optimal. Moreover, if there are no bicharacterisitcs tangent to $\mathrm{\Sigma }$, then the Cauchy problem is well-posed in the Gevrey class $1<s<3$ for all lower-order terms, and this result is also optimal. In the present paper, we remove the restriction on the codimension of $\mathrm{\Sigma }$, thereby completing the result.