Complexification of real k-theory gives index $2$ subgroup of complex k-theory

Question

We have $\widetilde{KO}(S^4) \cong \mathbb{Z}$ and $\widetilde{K}(S^4) \cong \mathbb{Z}$. There is a map $i:\widetilde{KO}(S^4) \rightarrow \widetilde{K}(S^4)$ that takes a stable vector bundle to its complexification. Question: What is the image of $i$? I've seen it asserted (say, in various sources discussing Rochlin's theorem on signatures of $4$-manifolds) that it is $2 \mathbb{Z}$, but I'm having trouble proving this.

Answer 1

This is the same as the induced map $\pi_4(KO) \to \pi_4(KU)$. Via Bott periodicity (see also this answer), this is the same as the induced map $\pi_0(KSp) \to \pi_0(KU)$ coming from the map sending a stable quaternionic vector bundle on a point to its underlying stable complex vector bundle relative to some choice of embedding $\mathbb{C} \to \mathbb{H}$. This map doubles dimensions.