Here is what I know about Whitehead products in homotopy groups of spheres:

 - $[\mathrm{id}_{S^{2n}},\mathrm{id}_{S^{2n}}]$ has Hopf invariant (EDIT: $\pm$) two.
 - No element that survives into the stable range can be a Whitehead product, since the suspension of a Whitehead product is trivial.
 - If $\alpha \in \pi_m(S^n)$ with $m$ odd, then $[\alpha,\alpha]$ has order at most 2.  This is because the Whitehead product is a graded Lie bracket.

I have two questions, one open-ended and one more specific.

**The open-ended question:** What are some known examples of nontrivial Whitehead products in finite homotopy groups of spheres?  Are there other situations, besides the ones mentioned, where Whitehead products must be trivial or have order at most 2?

**The specific question:** Is there an example of $\alpha \in \pi_m(S^n)$, for some $m>n$, such that $2\cdot[\Sigma\alpha,\Sigma\alpha] \neq 0$?