I am looking for a method to cut objects on ALL lines by thereselves. Hard to explain, here is an example of 4 objects (rects): http://www.weissarbeit.com/rect3213.png
On the left you see these 4 rects. Now I want to cut all lines in one step. The result is shown on the right side: 16 independet rects. How is this possible?

THX, fabske

I think you'll need two steps, first duplicate everything, select 2 green and 2 red ones and do Path > Division. Then select the remaining red ones and bring them to front (Home), add the green ones to selection and do a Division.

You'll get 4 extra squares on the intersections, you can delete them.

Thank you for you tipp, but there is one problem: I can only select one green and one red to do the division! If i select more objects, divison has no effect!?

Yes you have to go two by two, that's the only way Division knows what you want to do exactly.