Geometric proof of the fact that product of two elements of odd order in image of $J$ is zero.

Question

I want to prove that product of two elements of odd order in image of $J$ is zero. I tried to approach this via the Thom-Pontryagin condition. So the image of $J$ just corresponds to spheres with framing. Now I am not sure how to prove that the product is zero when the framings have odd order. So I am looking for a reference which does this in a geometric way. Thanks!