Characterization of complex line bundles induced from $\mathbb C P^1$

Question

Let $\eta$ be a complex line bundle over some (good) space. Then it is induced from the canonical line bundle over $\mathbb C P^{\infty}$. It may happen that $\eta$ in fact is induced from $\mathbb C P^1$ (that is, the classifying map is homotopic to a map into $\mathbb C P^1$). Then we have that $\eta \oplus \bar \eta$ is trivial (topologically) since it is true for any line bundle over projective line.

Question: is the converse true? Is it true that if for a line bundle $\eta$ we have that $\eta\oplus\bar\eta$ is trivial, then $\eta$ is induced from $\mathbb C P^1$?

Answer 1

Yes, if we have line bundles $\eta$ and $\zeta$ over $X$ and a trivialisation of $\eta\oplus\zeta$ given by a continuously varying family of isomorphisms $f_x\colon\eta_x\oplus\zeta_x\to\mathbb{C}^2$ then we can define $g\colon X\to\mathbb{C}P^1$ by $g(x)=f_x(\eta_x\oplus 0)$ and we find that $f$ induces an isomorphism from $\zeta$ to $g^*(\text{tautological bundle})$. More generally, a line bundle $\eta$ over $X$ can be classified by a map $X\to\mathbb{C}P^n$ iff $\eta$ can be embedded in a trivial bundle of dimension $n+1$.