TITLE: What is the generator of $\pi_9(S^3)$?
QUESTION [19 upvotes]: I have Toda's book Composition methods in homotopy groups of sphere.
I have not found the generator of $\pi_9(S^3)=\mathbb{Z}_3$.
What is it?

REPLY [10 votes]: Here's another way to describe the generator of $\pi_9(S^3)\cong\mathbb{Z}/3$.
By Proposition 13.6 in Toda's book, and the remarks preceding it, a generator is given by the composition
$$
S^9 \stackrel{\alpha_1(6)}{\longrightarrow} S^6 \stackrel{\alpha_1(3)}{\longrightarrow} S^3
$$
where $\alpha_1(3)\in \pi_6(S^3)\cong\mathbb{Z}_{12}$ is a generator of the $3$-primary component and $\alpha_1(6)=E^3\alpha_1(3) \in \pi_9(S^6)\cong\mathbb{Z}_{24}$ is its third suspension. 
The element $\alpha_1(3)$ has a very nice geometric representative described in my answer here. Explicitly, let $\mathbb{H}$ denote the quaternions, and represent the $6$-sphere as
$$
S^6 = \{(p,w)\in \mathbb{H}\times\mathbb{H} \mid \mathfrak{Re}(p)=0\mbox{ and } |p|^2+|w|^2=1\}.
$$
The map $b:S^6\to S^3\subseteq \mathbb{H}$ is given by 
$$
b(p,w) = \left\{\begin{array}{ll} \frac{w}{|w|} e^{\pi p} \frac{\overline w}{|w|}, & w\neq 0 \\
-1, & w=0, \end{array}\right.  
$$
where $e^{\pi p} = \cos(\pi |p|) + \sin(\pi|p|) \dfrac{p}{|p|}$ is the quaternionic exponential.  
This description appears in the paper
Abresch, U.; Durán, C.; Püttmann, T.; Rigas, A., Wiedersehen metrics and exotic involutions of Euclidean spheres, J. Reine Angew. Math. 605, 1-21 (2007). ZBL1125.57017.