In Selick's very pretty paper "Odd primary torsion in $\pi_*(S^3)$" he makes use of an automorphism which was established by Toda in his paper "On the double suspension $E^2$".

Unfortunately, Selick just refers to [10], but the paper is about 40 pages and dense reading; after skimming it several times, I fear that the automorphism is there either implicitly, or with different notation.

Can someone provide a more precise reference? A page or theorem number would be ideal.