Consider the J-homomorphism $\pi_{n-1}(SO(n))\to \pi_{2n-1}(S^n)$ and $\tau_{S^n}:S^{n-1}\to SO(n)$ to be the adjoint of the classifying map of the tangent bundle of the standard sphere. In this paper, Wall claims that $J(\tau_{S^n})=[id_n,id_n]$ where $id_n$ is the identity map of $S^n$. How can one see this?