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.)