Automatic timers for 4-way switch, or how to automatically turn off lights in a stairwell
Without any actual wiring diagrams or a picture of the wiring itself I would suggest a light with a presence sensor that has a timer inside. 
Almost all lamps of this type turn on when power is applied to them, even though they don't sense anyone.
Set the timer inside them to 10 minutes and when you switch the light on it will open. After 10 minutes of inactivity it will shutdown. If the lights are switched off and and the sensor does not 'see' you, simply switch the light twice. Once for power off then for power on.

If you post a picture with the wiring, we might come with a better solution.