You only pay for the bike in an S&S case (26x26x10 inches, or about 158 cm total) if it is overweight, or beyond your free luggage allowance (usually one piece of baggage, but maybe more if you are preferred status or whatever). Personally, I use S&S soft backpack cases and pack my bike and just about all my clothes in the case, making sure it doesn't go over 50 pounds (23kg). Getting a small digital luggage scale and bringing it with you is an easy way to max out your weight allowance (last time I flew with my S&S bike, my case weighed 49.9 pounds - I was proud and the check-in person was impressed!).

Lufthansa follows these guidelines, for example (see Free baggage rules on Lufthansa - Lufthansa ® United States of America). I assume their Europe rules are the same.