Equivalence between two definitions of $J$-homomorphisms.

Question

For a while, I have been studying $J$-homomorphism : $J_{k,n}: \pi_{k}{SO(n)} \rightarrow \pi_{n+k}S^{n}$. The most common definition is given using Hopf construction. However, there is an equivalent definition as the following composition: Consider the map $\phi:SO(n)\to M_*(S^n,S^n)\cong \Omega^nS^n$ in which an element $f:\mathbb{R}^n\to\mathbb{R}^n$ of $SO(n)$ is sent to a based map $\widehat f:S^n\to S^n$ by taking one-point compactifications (and where the basepoint is $\infty$). Then we can equivalently define the $J$-homomorphism as the map $\phi_*:\pi_k(SO(n))\to\pi_k(\Omega^n S^n)\cong\pi_{n+k}(S^n)$ induced by $\phi$. I have seen people using both definitions interchangeably. In this paper https://www.jstor.org/stable/pdf/1969085.pdf on page 473 the author claims something similar (He denotes $J$-homomorphism by $H$ and the compostion is $H = I_{\iota}JL$). Can someone please help me find the equivalence between two definitons? I am trying a lot but can not seem to get it. Thank you.