Classification of Quadratic Harmonic Maps of S^7 into S^7

complete classification of full quadratic harmonic maps of the seven-sphere S^7 into itself is obtained. It is shown that any full quadratic harmonic map of the seven-sphere S^7 into itself is equivalent to either the gradient map of the cubic isoparametric polynomial of É. Cartan or a map depending only on one parameter. The components of the map in the latter case are explicitly given.