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Let $\varPhi$ be a root system of a finite Weyl group $W$ with simple roots$\Delta$ and corresponding simple reflections $S$. For $J \subseteq S$, denoteby $W_J$ the standard parabolic subgroup of $W$ generated by $J$, and by$\Delta_J \subseteq \Delta$ the subset corresponding to $J$. We show that thelongest element of $W$ is decomposed into a product of several ($\le |\Delta|$)reflections corresponding to mutually orthogonal roots, each of which is eitherthe highest root of some subset $\Delta_J \subseteq \Delta$ or is a simpleroot. For each type of the root system, the factors of the specifieddecomposition are listed. The relationship between the longest elements ofdifferent types is found out. The uniqueness of the considered decomposition isshown. It turns out that subsets of highest roots, which give the decompositionof longest elements in the Weyl group, coincide with the cascade of orthogonalroots constructed by B.Kostant and A.Joseph for calculations in the universalenveloping algebra.

Decomposition of the longest element of the Weyl group using factors corresponding to the highest roots