  The fusion procedure of dilute $A_L$ models is constructed. It has been shown
that the fusion rules have two types: $su(2)$ and $su(3)$. This paper is
concerned with the $su(2)$ fusion rule mainly and the corresponding functional
relations of commuting transfer matrices in the $su(2)$ fusion hierarchy are
found. Specially, it has been found that the fusion hierarchy does not close.
These two types of fusion generate different solvable models, but, they are not
totally irrelevant. The $su(2)$ fusion of level $2$ is equivalent to the
$su(3)$ fusion of level $(1,1)$. According to this relationship the Bethe
ansatz of fused model of level $(1,1)$ in $su(3)$ hierarchy has been
represented by that of level $2$ in $su(2)$ fusion hierarchy.
