Most airlines simply choose to restrict all such items.
Apart from that list, every airline is free to set it's own policy with respect to what else can or can not be carried on board. Nearly all commercial airlines allow a rig as either a carry-on or checked item. In practice, very few skydivers are being denied boarding with their AAD-equipped rigs. Here are traveling tips. For starters, pack your rig in a gear bag or piece of luggage that will fit the size requirements for the airline.