The device reports every few days a motherboard failure. Toner, Drum and Error lights at the same time. Recommended replacement of the device as described by Brother unfortunately nothing has brought. As I feared, the alleged motherboard error is rather the expression of a software problem. After an off-on cycle of the error comes back sooner or later.
I do not know what the trigger is exactly. I have Google Cloud Print and AirPrint Secure in use. The printer depends on the LAN to a Fritz box. Routers and printers with the most current firmware.