  Let $\A$ be an irreducible Coxeter arrangement and $\bfk$ be a multiplicity
of $\A$. We study the derivation module $D(\A, \bfk)$. Any two-dimensional
irreducible Coxeter arrangement with even number of lines is decomposed into
two orbits under the action of the Coxeter group. In this paper, we will
{explicitly} construct a basis for $D(\A, \bfk)$ assuming $\bfk$ is constant on
each orbit. Consequently we will determine the exponents of $(\A, \bfk)$ under
this assumption. For this purpose we develop a theory of universal derivations
and introduce a map to deal with our exceptional cases.
