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I have one of these https://www.vincita.co.th/collections/brompton-bags/products/b132b-soft-transport-bag-for-brompton-bike and tend to borrow one of these https://www.vincita.co.th/collections/brompton-bags/products/b206b-garment-bag-for-b132b (although I'll be buying one when I get round to it). They're OK although the airline did break it last time it was used (but accepted responsibility and replaced with a new bag). The bag comes easily in normal checked suitcase size for most airlines so doesn't need to go as sports goods/oversized, etc.
Stick the brompton in folded, a bit of bubble wrap on some of the protruding areas, use the garment bag filled with clothes to pad out the top and sides and there is also space to slide a couple of bags with clothes, etc down the sides. I normally go with a C bag down one side and a rucksack down the other.
SJS have the hard version of that bag available https://www.sjscycles.co.uk/transporting-bicycles/vincita-b132c-transport-box-for-brompton/
how does one fly with brompton? I want n00b's guide.