I am trying to proove that $\pi_{n+1}(S^n) \cong \mathbb{Z}_2$ using the Pontryagin-Thom construction and the special case $n=1$ of the $J$-homomorphism $$ J_1:\pi_1(SO(n))\rightarrow \pi_{n+1}(S^n). $$ but I can't find any reference of this specific case of the $J$-homomorphism being an isomorphism.
Where can I find a proof?