  In 1990, D.Snow proposed an effective algorithm for computing the orbits of
finite Weyl groups. Snow's algorithm is designed for computation of weights,
$W$-orbits and elements of the Weyl group. An extension of Snow's algorithm is
proposed, which allows to find pairs of mutually inverse elements together with
the calculation of $W$-orbits in the same runtime cycle. This simplifies the
calculation of conjugacy classes in the Weyl group. As an example, the complete
list of elements of the Weyl group $W(D_4)$ obtained using the extended Snow's
algorithm is given. The elements of $W(D_4)$ are specified in two ways: as
reduced expressions and as matrices of the faithful representation. We present
a partition of this group into conjugacy classes with elements specified as
reduced expressions. Various forms are given for representatives of the
conjugacy classes of $W(D_4)$: using Carter diagrams, using reduced expressions
and using signed cycle-types. In the appendix, we provide an implementation of
the algorithm in Python.
