  In this paper, we pursue the investigations started in \cite{Mas-You} where
the authors provide a construction of the Dunkl intertwining operator for a
large subset of the set of regular multiplicity values. More precisely, we make
concrete the action of this operator on homogeneous polynomials when the root
system is of dihedral type and under a mild assumption on the multiplicity
function. In particular, we obtain a formula for the corresponding Dunkl kernel
and another representation of the generalized Bessel function already derived
in \cite{Demni0}. When the multiplicity function is everywhere constant, our
computations give a solution to the problem of counting the number of
decompositions of an element from a dihedral group into a fixed number of (non
necessarily simple) reflections. In the remainder of the paper, we supply
another method to derive the Dunkl kernel associated with dihedral systems from
the corresponding generalized Bessel function. This time, we use the shift
principle together with multiple combinations of Dunkl operators corresponding
to the vectors of the canonical basis of $\mathbb{R}^2$. When the dihedral
system is of order six and only in this case, a single combination suffices to
get the Dunkl kernel and agrees up to an isomorphism with the formula recently
obtained by Amri \cite[Lemma1]{Amri} in the case of a root system of type
$A_2$. We finally derive an integral representation for the Dunkl kernel
associated with the dihedral system of order eight.
