No, it is used stand alone
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After thinking more about your (very relevant) question, I thought about a problem with the internal monitor indeed.
Mechanics & electics appear to be less reliable than solid state things, and I went to suspect something about the screen. I actually found a way to restart the computer by bending the screen in the lower right corner near the on/off button.
Now the problem seem to be how to check and possibly fix a connexion problem with the screen?
Thank you very much for this insight, we actually made progress.
...
After further tries I am no longer sure it is the actual issue. Coincidence seems the actual link between the succesful start and the screen bent.
So the issue is still: the laptop does not start, the screen stays black, though it sometimes - less and less often - starts and then works normally.