  This note gives a uniform, self-contained, and fairly direct approach to a
variety of obstruction-theoretic problems on 8-manifolds. We give necessary and
sufficient cohomological critera for the existence of almost complex and almost
quaternionic structures on the tangent bundle and for the reduction of the
structure group to U(3) by the homomorphism U(3) --> O(8) given by the Lie
algebra representation of PU(3).
