Take off the bottle cages, mounting bolts back into the frame.
Take off racks, mounting bolts back into the frame.
Loosen but do not remove stem and beam.
Turn the bike upside down, onto bars and seat.
Take off pedals, remember, one goes the wrong way.
Open the brakes and take off the wheels and let out the air.
Take out the skewers and put them in your small parts bag with pedals.
Fold while upside down and slip frame into suitcase. Front facing left and seat in your stomach.
Remove (loosened) stem and remove and rejoin handlebars.
Remove (loosened) beam and remove seat & post.
Frame now slides all the way into suitcase.
After frame is in suitcase shift rear derailleur to low (ie inward) to protect it.
Insert other elements as shown.
Rear rack lies across top, in its bag, making hole in bag for crush protector.
Bottle cages, pedals, tools, even shoes and helmet can all go into the empty spaces.

Bike is now luggage and flys safely and without charge.
Bike bag will roll up into a stuff bag and can be strapped beneath the rear rack over the tire.

Many European trains now take bikes and a few American ones do but with the bag you can travel on trains or buses anywhere, anytime with your bike regardless of regulations. This is not difficult. Many, many bikers do this every year. Your bike will come apart like Tinkertoys and go back together the same way. Just be slow, observant and thoughtful.
NB Leftover bits are not good. P>