  A Coxeter group of classical type $A_n$, $B_n$ or $D_n$ contains a chain of
subgroups of the same type. We show that intersections of conjugates of these
subgroups are again of the same type, and make precise in which sense and to
what extent this property is exclusive to the classical types of Coxeter
groups. As the main tool for the proof we use Solomon's descent algebra. Using
Stirling numbers, we express certain basis elements of the descent algebra as
polynomials and derive explicit multiplication formulas for a commutative
subalgebra of the descent algebra.
