TWhat you have pictured there Markus, is not the demi besaigue, but a similar tool

Demi Besaigue is the French name for a German tool, which is known as a Stichaxt or Stossaxt. Sometimes this is in English called a push axe. It occupies a gray area somewhere between the axe and the chisel. It is often referred to as the 'German Slick' and the Germans refer to the slick as being an American or British thing.

http://www.traditionalwoodworker.com/1-3...tinfo/598-4500/
There is one you can get in the US, made by Ochsenkopf

I have one, and I love it. It is useful for a number of things, and is perhaps more versatile than the slick. Traditionally it is used in Germany as an all purpose cleanup tool. Used to clean mortises and other such joints like a slick, but also used to smooth timber faces -instead of an adze or a variety of hand planes- which it works rather well at.

The French Besaigue is a development of the German Stichaxt (which itself is a variation of the older Kreuzaxt, which in one form is more or less what you have linked to there.) in which a heavy duty mortising chisel has been added at the other end. both tools are essentially used the same way.

To use the stichaxt, you need to have the timber fairly low to the ground, so that you stand mostly above it. You will see in old pictures from Germany and France that the timbers are always low to the ground, never worked high up on horses. This is necessary because you use your body weight to work the stichaxt.

You work in a kneeling position with one hand on the tools short handle and the other a few inches down on the body, and with a rocking motion push the tool across the wood. It can make very fast work of removing relatively large quantities of wood.

The Kreuzaxt from which it was developed is essentially a twibil, and was originally used to rough out mortises that had first been bored out. eventually they just decided to take the wooden handle off of it. It is not clear what all it was used ofr originally, but eventually it came to be used as a cleanup tool.

The Besaigue may or may not have been originally French, as I understand it is generally considered to have been invented somewhere in or around Alsace, a historically German speaking region that is part of France today.

It is important to note that the Germans have a specific system of tools worked out for timber framing. They are well adapted to their style of joinery, and perhaps more importantly well adapted to each other. The Stichaxt is just one of these tools.
If It were an american tool, we would probably call it a chisel, but the Germans mostly used axes to do things that we would use chisels and saws for, and so they saw it as one of a series of axes used to cut a frame. hence the name Stichaxt (Axt being one word in German for axe)