Since my old answer is referenced here What is the generator of $\pi_9(S^3)$?, I spent a little time trying to figure out what it says about this. "Toda's sequence" is a $p$-local fiber sequence ($p$ any prime) of the form
$$ S^{2n-1} \to \Omega \widehat{S}^{2n} \to \Omega S^{2pn-1},
$$
where $\widehat{S}^{2n}$ is a certain space I don't need to care about yet; see https://en.wikipedia.org/wiki/EHP_spectral_sequence. This backs up one step to a $p$-local fiber sequence
$$
\Omega^2 S^{2pn-1} \xrightarrow{f} S^{2n-1} \to \Omega \widehat{S}^{2n}.
$$
The bottom non-trivial homotopy group of the fiber is $\mathbb{Z}$, and so we get a map
$$
\mathbb{Z}=\pi_{2pn-1}S^{2pn-1}=\pi_{2pn-3} \Omega^2 S^{2pn-1} \xrightarrow{f_*} \pi_{2pn-3}S^{2n-1}.
$$
When $p=3$ and $n=2$ this gives $\mathbb{Z}=\pi_{9}\Omega^2S^{11}\to \pi_9S^3$, which by the argument I gave in my other answer surjects with image $\mathbb{Z}/3$. So we want to compute the image of $f_*$ in this dimension, i.e., the effect on $f$ on the bottom cell of $\Omega^2 S^{2pn-1}$.

Now, when $p=2$, "Toda's sequence" is the James's EHP sequence (with $\widehat{S}^{2n}=S^{2n}$). In this case $f_*$ is $\mathbb{Z}\to \pi_{4n-3}S^{2n-1}$. We know what the image of the generator is in this case: it is the "Whitehead square" $[\iota_{2n-1}, \iota_{2n-1}]$, which can be described geometrically as the attaching map of the $4n+2$-cell of $S^{2n-1}\times S^{2n-1}$.

I then tried to figure out what Toda's sequence actually is, in hopes of finding a geometric description of the map, and I failed. Everyone seems to refer to Toda's *Composition Methods* book for this sequence, but the statement given on Wikipedia is not actually there. However, the idea of the sequence (together with the other Toda sequence described on wikipedia) seems to be essentially 13.1 in Toda's book. In turn, 13.1 is not proved in Toda's book, but rather in the paper

Toda, Hirosi: On the double suspension $E^2$, J. Inst. Polytech. Osaka City Univ. Ser. A. 7 (1956), 103–145.

The fiber sequence I am interested in is essentially Theorem 7.6 of Toda's paper.

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