Not sure from your posting whether you are losing a "wireless" connection between computer "A" and the Livebox, or the ADSL connection between the Livebox and the internet.
I have heard of similar instances where an unconnected computer has caused the loss of an ADSL signal and can only suggest that some (unknown) component in computer "B" is emitting electromagnetic radiation at a frequency which interferes with the ADSL signal.
There are some really strange things that can affect an ADSL signal. One of the most bizarre I've come across was roadworks where the phone lines were exposed and the temporary traffic lights effectively killed the ADSL signal. In another case, someone lost internet access at dusk every night. It turned out to be a street light coming on close to overhead phone lines killed the ADSL signal.