What is the proper technique to self-arrest and brake a fall, given steep icy (frozen snow) slope, when wearing crampons and wielding an ice axe? Should the crampons be used for stopping or held high in the air, to avoid tumbling over them?
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You never want to stop yourself with the crampons because they are liable to catch, flip you over and, at best, put you in a worse situation than before and, at worst, break your legs.
Instead you want to first stop yourself using the pick of the ice axe, with your crampons raised above the ice. You can use your knees as an additional brake. The way you do this is first get into self arrest position, which is face down, with legs downhill, one hand on the head of the axe, and one hand down the shaft, with the head on level with the shoulder. The pick is placed into the snow, and the bottom of the shaft raised, with the bulk of the torso placed on the shaft and and pick.
After you are stopped, or sufficiently slowed down, use the crampons to kick steps into the ice, and get yourself into a secure position.
It is highly recommended to practice this with knowledgeable mountaineering guides or instructors on relatively safe slopes.
Although @ReverendGonzo gave a nice answer I want to start a little debate. There is no explicit answer to this question. Different alpine clubs have different opinions and even different mountain guides in one organization. That being said, I think the process described by @ReverendGonzo (which I will call default process) is very common and also the doctrine of the biggest German Alpine club ("Deutscher Alpenverein").
Still, I will share the opinion of a very experienced guide* I was pleased to have a course with. The question was about a steep slope with frozen snow. That isn't ice. For bare ice holding a fall is even more troublesome. If you are on snow (which is most likely for a classical Alpine tour), his suggestions were:
Those previous thoughts are all together just a point of view against the default process. In theory this default process works but may be really difficult under real circumstances. Therefore the suggestion was to get you self-arrest really fast or it's very unlikely at all (if you're not roped in). Get yourself in a pushup position, use your hands/fists (always wear gloves!), crampons, maybe also the knees.
That being said I don't want to state that this point of view is better or describes the best strategy at all. The specific conditions of the snow, weather, equipment, the degree of the slope, security possibilities, ... all that has an impact. I was a bit surprised by the opinions of the guide but this somehow opened my eyes. I wanted to share this with you.
There are no rules.
*He was raised directly in the Alps close to big mountains, working as a professional mountain guide and ski guide for nearly 30 years. He also works for the mountain rescue team.