I ended up doing the job with a jigsaw and a drill. Jigsaw saws are available in several sizes. One of them went exactly 10cm deep. Of course it would bang into the wood if I would saw the 3 plains. To minimize the banging I drilled 3 holes along the end of those plains. I hope these pictures help to make sense out of this.
Keep the jigsaw firm and progress it slowly forward. The drilled holes might be off a bit and the saw will bang in the wood as it progresses. Need to keep the jigsaw pressed against the wood to absorb the mild blows.
Here's the end result after shaving a bit with a wood file.
Oh and the 2cm that's left is strong enough. That won't break easily. Not with your bare hands or fingers at least as someone suggested.
This method will probably not work in hardwood.