triviality of tensor product of vector bundles Let $\xi$ be a $O(n)$-bundle with fibre $\mathbb{R}^n$. Let $\xi\otimes \mathbb{C}$, $\xi\otimes \mathbb{H}$ be complex vector bundles and quaternionic vector bundles. If $\xi$ is not a trivial bundle, can we obtain $\xi\otimes \mathbb{C}$, $\xi\otimes \mathbb{H}$ are not trivial bundles?

The Möbius real line bundle bundle $\xi$ over the circle $S^1$ is not trivial but its complexification $\xi\otimes_\mathbb R \mathbb C$ is trivial, like all complex line bundles over $S^1$.
[This last fact is due to complex line bundles on the circle being classified by $H^2(S^1,\mathbb Z)=0$]