How do I show that $\pi_* (\mathbb{S}(\mathbb{T})) = \pi_* \mathbb{S}[d]/(d^2 - \eta d)$? In http://web.math.ku.dk/~larsh/papers/s07/handbook.pdf they say The relation $d^2 = \eta d$ is a
consequence of the fact that, stably, the multiplication map $\mu: \mathbb{T}\times \mathbb{T} \to \mathbb{T}$ splits off the Hopf map, but I don't know what that means :(