triviality of tensor product of vector bundles

Question

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?

Answer 1

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 $H^2(S^1,\mathbb Z)=0$]

Answer 2

The tangent Bundle of sphere is another example. For arbitrary vector bundle E, it seems that $E\otimes \mathbb{C}$ has a trivial line bundle, as a summand.

https://mathoverflow.net/questions/209247/almost-complex-structure-and-nontrivial-idempotents

Moreover, I guess that the complexification of canonical line bundle over $\mathbb{R}P^{n}\;\;n>1$ is nontrivial.