So recall that if I have a map $f : S^{4n-1} \to S^{2n}$ the Hopf invariant $h(f)$ of $f$ is defined in the following way. Letting $X = S^{2n} \cup_f e^{4n}$ there exists a short exact sequence $$0 \to \widetilde{K}(S^{4n}) \xrightarrow{g^*} \widetilde{K}(X) \xrightarrow{i^*} \widetilde{K}(S^{2n}) \to 0.$$ Now recall that $\widetilde{K}(S^{4n}) = \mathbb{Z}$ so let $b \in \widetilde{K}(X)$ be the image of the generator $i_{4n}$ of $\widetilde{K}(S^{4n})$ and let $a \in \widetilde{K}(X)$ be the element that maps to the generator $i_{2n}$ of $\widetilde{K}(S^{2n})$, in other words $i^*(a) = i_{2n}\in\widetilde{K}(S^{2n})$. Then $\widetilde{K}(X)$ is the free abelian group on the basis $\{a, b\}$.
Then to show that $h(f) = \pm 1 \implies n = 1, 2$ or $4$ one uses the Adams operations and this is where my question comes in.
Using the naturality of the Adams operations $\psi^k$ and the fact that for any non-zero integer $k$ we have $\psi^k(u) = k^m u$ if $u \in \widetilde{K}(S^{2m})$ it follows that $$\psi^k(b) = \psi^k(g^*(i_{4n})) = g^*(\psi^k(i_{4n})) = g^*(k^{2n}i_{4n}) = k^{2n}g^*(i_{4n}) = k^{2n}b.$$
Now usually next authors claim that $$\psi^k(a) = k^na + \mu_kb$$ for some $\mu_k \in \mathbb{Z}$ and this is what I don't understand. How does one arrive at the above equation?
This is my attempt to show the above. Since $\psi^k(a) \in \widetilde{K}(X)$ we have that $\psi^k(a) = \eta_k a + \mu_k b$ for some integers $\eta_k, \mu_k$. Using naturality of the Adams operations and the fact that for any non-zero integer $k$ we have $\psi^k(u) = k^m u$ if $u \in \widetilde{K}(S^{2m})$ it follows that $$i^*(\psi^k(a)) = \psi^k(i^*(a)) = \psi^k(i_{2n}) = k^n i_{2n}.$$ But also $i^*$ is a ring homomorphism and hence $i^*(\psi^k(a)) = i^*(\eta_k a + \mu_k b) = \eta_ki^*(a) + \mu_ki^*(b)$ thus we have $$\eta_ki_{2n} + \mu_ki^*(b) = k^n i_{2n}.$$
But if I show that $\eta_k = k^n$ then it would imply that $\mu_k = 0$ so at this point I'm not sure how to proceed. I'm not sure whether the claimed equation arises just from applying standard algebraic arguments or if there's some underlying topological reason that I can't see at the moment. It's also possible I might be making some silly algebraic mistake but I can't see that at the moment either.