TITLE: Simplest example of non-trivial Toda bracket in spaces
QUESTION [11 upvotes]: Many sources give an easy definition of a Toda bracket $\{f,g,h\}$ of appropriate maps $W \to X \to Y \to Z$ in spaces as a subset of the homotopy classes of maps $[\Sigma W, Z]$ (for example, Ravenel's Complex Cobordism.)  However, the only real usages of Toda brackets that I can find are in the context of heavier lifting, in what appear to be difficult computations of stable homotopy groups of spheres.
It's not difficult to construct maps satisfying the nullhomotopy criteria, but either it's not clear how to work things out or I wind up finding only a trivial bracket.
Some sources give more conceptual accounts (such as this paper), but again the only examples are quite complicated from my inexperienced point of view.  For instance, the only example in the linked paper is an element in $\pi_{11}(S^6)$ constructed from maps I'm not familiar with.
What's the simplest natural example of a non-trivial Toda bracket in spaces?  It would be especially helpful if the example illustrated the intuition or geometry behind Toda brackets, but I understand that may not be how things are.

REPLY [7 votes]: Just wanted to expand @Achim Krause 's final comment: Mosher and Tangora has a nice treatment that ramps up from more to less elementary (they do get to statements in terms of Toda brackets eventually).  They work stably by considering $S^n$ for $n$ large, and include the following examples involving the (suspensions of) Hopf maps $\eta, \nu, \sigma$:

Assume $\eta^2 = 0$; then use part of the bracket construction and the relation $Sq^2Sq^2 = 0$ to derive a contradiction.
Similar strategy to show $\eta^3 = 4 \nu$
$\eta^2 \in \langle 2, \eta, 2 \rangle$
$2 \nu  \in \langle \eta, 2, \eta \rangle$
$\nu^2 \in \langle \eta, \nu, \eta \rangle$
$8 \sigma \in \langle\nu, 8, \nu\rangle$