The topological obstructions to the existence of an irreducible SO(3)   structure on a five manifold
  A nonstandard (maximal) inclusion SO(3) in SO(5) associated with the irreducible representation \rho_5 of SO(3) in R^5 is considered. The topological obstructions for admitting the SO(3) structure on the frame bundle over 5-manifold are investigated. The necessary and sufficient conditions are formulated. 
Marcin Bobienski