Odd K-theory of the sphere bundle of two tautological lines ($K^{-1}(S(\gamma \oplus \gamma ))$)

For $E=\gamma \oplus \gamma$ over ${\mathbb{CP}}^n$ and $t=1-[\overline{\gamma }]$, the K-theory Euler class is $t^2$. Hencefor $n\ge 1$.

For $n=0$, the base is a point, the sphere bundle is $S^3$, and its odd K-theory is $\mathbb{Z}$.