  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.
