In the paper "Clifford modules" by Atiyah, Bott and Shapiro, a homomorphism $\alpha:A_k\rightarrow \tilde{KO}(S^k)$ from a certain group of Clifford modules to real $K$-theory of spheres is constructed. The composite of this homomorphism with the stable $J$-homomorphism $J:\tilde{KO}(S^k)\rightarrow\pi_{k-1}^s$ gives a homomorphism $J\circ\alpha:\alpha:A_k\rightarrow\pi_{k-1}^s$ from $A_k$ to stable homotopy groups of spheres.
My question is the following: Is there a direct homomorphism $\beta : A_k\rightarrow \rightarrow\pi_{k-1}^s$ from Clifford modules to stable homotopy without mentioning vector bundles and $J$-homomorphism in the literature?
Any hint or idea about this question and reference suggestions about these topics would be greatly appreciated.
