Local-global principle for triangularizability and diagonalizability of matrices

Given a number field $k$ with the ring of integers ${\mathcal{O}}_k$ and a matrix $M\in {\mathrm{M}}_n({\mathcal{O}}_k)$. We prove that if ${\mathcal{O}}_k$ is a principal ideal domain, the local-global principle for triangularizability and diagonalizability of $M$ holds. To explain the possible failures of the local-global principle, we prove that the stratified Brauer–Manin obstruction is the only obstruction to the local-global principle for triangularizability and diagonalizability of $M$ in some special cases.