Can somebody say something about surjectivity of the $J$ homomorphism $J_{7,7}$ : $\pi_{7}$SO($7$) $\rightarrow$ $\pi_{14}(S^7)$ ? Husemoller in Fibre Bundles says this :
Where $M_{n}^{1}$ denotes component of $\text{Map}_{0}(S^{n-1},S^{n-1})$ consisting of maps of degree $1$. I do not know how to proceed after this. Also is there any information about unstable $J$ homomorphisms ?