A note on the topology of irreducible $\mathrm{S}\mathrm{O}(3)$-manifolds

We give necessary and sufficient topological conditions for the existence of an irreducible $\mathrm{S}\mathrm{O}(3)$-structure on a $5$-manifold. Using these conditions we provide some new examples of $5$-manifolds with an irreducible $\mathrm{S}\mathrm{O}(3)$-structure.