Weyl discriminant and reflecting hyperplane arrangement

Two nonzero root forms have the same complex kernel only if they are proportional. Restricting to real vectors makes the proportionality scalar real; reducedness then makes their roots equal up to sign, and positivity selects the same root. Every root reflection fixes its corresponding real hyperplane and its complexification. Conversely let $w\in W$ be a complex reflection, meaning $w\mathrm{\ne }1$ and its fixed complex subspace has codimension one. Since its matrix is real, its real fixed subspace has real codimension one too. Orthogonality forces $w$ to be the unique orthogonal reflection in that hyperplane $H$. If $H$ were different from every real root hyperplane, their intersections with $H$ would be finitely many proper subspaces of $H$. They cannot cover $H$: in a finite basis of $H$, substitute $(1,z,\dots ,z^{\dim H-1})$ in each of the nonzero restricted forms, and choose a real value outside the finite set of roots of those polynomials. In dimension zero, the assertion that $H$ differs from every root hyperplane can occur only if there are no roots in a positive-dimensional spanning root system, which is excluded. Thus $H$ would contain a regular point fixed by $w$, contrary to F2. Hence $H$ is a root hyperplane.

Orthogonality gives $w\cdot {\mathrm{\ell }}_{\alpha }={\mathrm{\ell }}_{w\alpha }$. By F1, applying $s_i$ to the product defining $\mathrm{\Delta }$ permutes every factor except ${\mathrm{\ell }}_{{\alpha }_i}$, which changes sign. Thus $s_i\cdot \mathrm{\Delta }=-\mathrm{\Delta }=\det (s_i)\mathrm{\Delta }$. Multiplication of these identities along any simple-reflection word yields $w\cdot \mathrm{\Delta }=\det (w)\mathrm{\Delta }$ for all $w$. No choice of a word for each group element is needed; the identity holds for every word. The polynomial is nonzero since its factors are nonzero in a polynomial ring over a field, as also follows by multiplying leading monomials in any fixed monomial order. Rank zero gives the same identity for the unit.