I'm wondering is that something common? Does it use at the competitions as well and what is the purpose of it? I've never seen anyone would do jumping lesson with the double bridle on until couple weeks ago, so was really surprised to see one.
I don't know if it's common, but I have no problem with it IF the rider knows how to properly ride a horse in a double bridle. Many people don't know how to and/or don't care to learn and they think they can just slap one on their horse and have a magical fix and everything will be fine. That's when bad things happen for the horse and then I do have a problem with it.
I really wonder if what you saw was truly a double bridle instead of a pelham. It's very hard to distinguish an all metal pelham from a double without examining it closely.
I personally wouldn't use a double on a horse used for jumping, but that's just me. I don't think the finesse possible with a double is going to be accessible while jumping a course. I have used a pelham, and would again if circumstances warranted.
The Hunter/Jumper Coach who comes out to the barn I am currently at - most of his students are in Pelhams. You can always decipher who his students are, in the show ring when you go to big LMHJA shows.
The issue I have, is that he *I can't speak for every coach* does not show his students how to use the reins properly and indavidually. His students go around, using both reins at the same time - always!
A pelham is a *single* leverage bit with a curb chain, ridden with two reins.
A double bridle actually has two separate bits, a snaffle, called a bridoon when it's part of a double, and a curb bit, called a weymouth (if I got that wrong, I hope a more knowledgable dressage rider will correct me) There are two reins, but there are actually two separate and distinct mouthpieces in the horse's mouth. It is required above a certain level in dressage (4th, I think.) It allows a great deal of finesse - for instance, you can influence collection with the curb while influencing direction with the snaffle. I have only ever seen them used on upper level dressage horses, but that doesn't mean someone isn't out there using them badly or incorrectly on other horses.
ETA: two reins does not equal a double. There are a bunch of leverage bits ridden with two reins: pelhams, three rings, elevators and some gags. Two *bits* = a double bridle.