Have flown with my Brompton to a few places; Hawaii, California, Puerto Rico, Japan, etc.
I usually just pack the bike in the original cardboard box; along with my clothes, tools, lights,
spares(tubes, tire, etc.), shoes, food, etc. And use the front luggage as my carry on - laptop,
snacks, electronics, etc. No problems usually.

One time I was going from New York to Los Angeles. Departed from JFK, then landed at LAX.
Threw the box at LA and rode to the place I'm staying in Anaheim. After a week I needed to
pack for my return trip. Just made a big box out of two smaller boxes:
Brompton folding bike; ready for check-in by 1nterceptor, on Flickr