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In 1990, D.Snow proposed an effective algorithm for computing the orbits offinite 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 isproposed, which allows to find pairs of mutually inverse elements together withthe calculation of $W$-orbits in the same runtime cycle. This simplifies thecalculation of conjugacy classes in the Weyl group. As an example, the completelist of elements of the Weyl group $W(D_4)$ obtained using the extended Snow'salgorithm is given. The elements of $W(D_4)$ are specified in two ways: asreduced expressions and as matrices of the faithful representation. We presenta partition of this group into conjugacy classes with elements specified asreduced expressions. Various forms are given for representatives of theconjugacy classes of $W(D_4)$: using Carter diagrams, using reduced expressionsand using signed cycle-types. In the appendix, we provide an implementation ofthe algorithm in Python.

Extending Snow's algorithm for computations in the finite Weyl groups