A note on the Hopf homomorphism of a Toda bracket and its application

The purpose of the present note is to extend a formula between the Toda bracket and Hopf homomorphism. As an application, we show that the generator of the 2-primary component of the homotopy group p12ðSÞ is taken as a representative of a specific Toda bracket. And we shall give a short proof of the existence of the unstable Adams map.