The Kac Moody denominator is Weyl skew

A positive root other than ${\alpha }_i$ has a positive simple coordinate at an index different from $i$: the only roots on the $i$th axis are $\pm {\alpha }_i$ by F1's root conventions. Reflection changes only coordinate $i$, and its image is a root by F2, so the one-sign property makes it positive. Applying the involution twice proves that $s_i$ permutes ${\mathrm{\Delta }}^{+}\setminus \{{\alpha }_i\}$, preserving every multiplicity. Also $s_i\rho =\rho -{\alpha }_i$ since $\rho (h_i)=1$.

Transform all exponents of the defining product. Step 1.1 gives $s_{i}D=e^{\rho -{\alpha }_i}(1-e^{{\alpha }_i}){\prod }_{\alpha >0,\alpha \mathrm{\ne }{\alpha }_i}(1-e^{-\alpha }{)}^{m_{\alpha }}=-D\mathrm{.}$ The positive-root product after reindexing is coefficientwise finite by F1; the single exceptional factor is a polynomial with two terms. Thus this manipulation really equals the transformed coefficient array, rather than assuming an action on the entire completion. A finite word of reflections now transforms this particular array repeatedly, producing one minus sign per reflection. Each simple reflection fixes a hyperplane and negates its complementary root line, so has determinant $-1$; the accumulated sign is $\det (w)$ independent of the word. The identity word has sign one. Every transformation used only finite coefficient computations, with no AC.