James proves a great number of things about the Whitehead product in his paper On the suspension sequence (though a number of results in that paper are stated in terms of cases rather than the stronger results that hold 2-locally). For example, he shows that there is a 2-local pairing
$$
\{-,-\}: \pi_p(S^k) \times \pi_q(S^k) \to \pi_{p+q+1}(S^{2k+1})
$$
such that $\{\alpha, \beta\} = (-1)^{pq + k} \{\beta, \alpha\}$ and such that the composite with the "Whitehead product" map
$$
P: \pi_{p+q+1}(S^{2k+1}) \to \pi_{p+q-1}(S^k)
$$
that appears in the EHP sequence is the ordinary Whitehead product. In particular, since the Whitehead product is graded-commutative we get an identity $2 [\alpha, \alpha] = 0$ whenever $k$ is odd, complementing the fact that $2[\alpha, \alpha] = 0$ by graded-commutativity whenever the source degree $p$ of $\alpha$ is odd. (In particular, this shows that when $n$ is even, $2 [\Sigma \alpha, \Sigma \alpha] = 0$ for any element $\alpha$ in the homotopy groups of $S^n$.)

James also shows naturality for the EHP sequence in a sense, and this has the following consequence. Writing the element $[\Sigma \alpha, \Sigma \alpha]$ as a composite of $[id_{m+1}, id_{m+1}]: S^{2m+1} \to S^{m+1}$ with the map $\Sigma \alpha: S^{m+1} \to S^{n+1}$, James' naturality shows that
$$
\{\Sigma \alpha, \Sigma \alpha\} = \Sigma^{m+1} \alpha \circ \Sigma^{n+1} \alpha \circ \{id_{m+1}, id_{m+1}\} = \pm 2 (\Sigma^{m+1} \alpha \circ \Sigma^{n+1} \alpha) = \pm 2 \Sigma^{n+1}(\Sigma^{m-n} \alpha \circ \alpha).
$$
and thus that
$$
[\Sigma \alpha, \Sigma \alpha] = \pm 2 P(\Sigma^{n+1} (\Sigma^{m-n} \alpha \circ \alpha)).
$$

In particular, for this element to not be 2-torsion, the element $4\Sigma^{n+1} (\Sigma^{m-n} \alpha \circ \alpha)$ must not be in the kernel of $P$. However, the EHP spectral sequence tells you that this is the same as asking that the element $4\Sigma^{n+1} (\Sigma^{m-n} \alpha \circ \alpha)$ must not be in the image of the Hopf invariant map $H$. (*Stably*, this element is $4 \alpha^2 = (2 \alpha)^2$.)

(This is the point where I say that I've exhausted most of my knowledge of unstable theory. I don't know if there are any examples of non-2-torsion self-Whitehead squares.)

existsif and only if $[\iota_{k+1},\iota_{k+1}]$ is divisible by 2? What do you mean by $\iota_{k+1}$ (a link is fine)? $\endgroup$3more comments