This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to the representation of some (classical) Lie group, I'd be very happy.
2026-03-25 07:40:43.1774424443
A smooth non-stably trivial smooth vector bundle
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The Möbius bundle over $S^1$ has this property.