This is more or less the same question as [ https://mathoverflow.net/questions/76899/what-is-the-generator-of-pi-9s3 ], except what I would like to know is if it is possible to describe this map in a way not only topologists can make sense of.

[EDIT] (Following the advice of Ryan Budney.) A purely geometric construction like the famous Hopf fibration for $\pi_3(S^2)$ would be perfect. (Something like a map $S^9\to S^2$ or $S^9\to S^3$ which may be written down in equations.) But I understand that there is little hope for that.

A less explicit but probably more reasonable approach is to try and represent this homotopy class by a framed 7-manifold in ${\mathbb R}^9$ following Pontryagin. In fact, any information about such a manifold may be of help. Is it really complicated?

I am not really familiar with the work of Jie Wu, but what I have read this far makes sense to me. So, the answer can also be along this lines, but if so I would like to see more then hints. (This computation looks horrendous, and I probably cannot handle it by myself.)