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How I can prove any element of order 2 in a Weyl group is the product of commuting root reflections. I need to show also that the only reflections in Weyl group are the root reflections.

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up vote 2 down vote accepted

Let $R\subseteq W$ be the set of reflections in the hyperplanes orthogonal to the roots. Of course, $R$ generates $W$, and there is then a function $\ell:W\to\mathbb N_0$ such that for each $w\in W$ the number $\ell(w)$ is the minimal length of an expression of $w$ as a product of elements of $R$. One can easily show that $\ell(w)$ is the number of eigenvalues of $w$ different from $1$ in the defining representation (in other words, the codimension of the subspace fixed by $w$)

From this it follows immediately that the only reflections are the elements of $R$.

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