Determination of the order of the P-image by Toda brackets

The present paper gives a proof of the author's paper in the Proceeding of the International Conference on Homotopy Theory and Related Topics at Korea University (2005), 109--113, on the orders of Whitehead products of iota_n with alpha in pi^n_{n+k}, (n > k+1, k < 25) and improves and extends it. The method is to use composition methods in the homotopy groups of spheres and rotation groups.