Almost quaternion structures of the second kind and almost tangent structures

is called an almost quaternion structure and a differentiate manifold with an almost quaternion structure an almost quaternion manifold. If there exists, in an almost quaternion manifold, a system of coordinate neighborhoods with respect to which components of F, G and H are all constant, then the almost quaternion structure is said to be integrable and the almost quaternion manifold is called a quaternion manifold. In a previous paper [8], the present authors studied integrability conditions for almost quaternion structures. A set of three tensor fields F, G and PI of type (1, 1) in a differentiate manifold which satisfy