Bruhat intervals of length 4 in Weyl groups

We determine all isomorphism classes of intervals of length 4 in the Bruhat order on the Weyl groups A4, B4, D4 and F4. It turns out that there are 24 of them (some of which are dual to each other). Work of Dyer allows us to conclude that these are the only intervals of length 4 that can occur in the Bruhat order on any Weyl group. We also determine the intervals that arise already in the smaller classes of simply laced Weyl groups and symmetric groups. thod combines theoretical arguments and computer calculations. We also present an independent, completely computerized, approach.