Affine denominator separates real and imaginary root factors

F3 and F4 give real roots $\alpha +n\delta$ for every finite root $\alpha$ and integer $n$, with multiplicity one. Their positive members are those with $n>0$, together with $n=0$ and $\alpha \in {\mathrm{\Phi }}_0^{+}$. For $n>0$ and $\alpha \in {\mathrm{\Phi }}_0^{+}$, F5 gives $\alpha +n\delta =n{\alpha }_0+(n\theta +\alpha )\in Q_{+}\mathrm{.}$ For $\alpha =-\beta$ with $\beta \in {\mathrm{\Phi }}_0^{+}$, it gives $-\beta +n\delta =n{\alpha }_0+(n-1)\theta +(\theta -\beta )\in Q_{+}\mathrm{.}$ These exhaust the two finite-root signs by F6. Negation handles $n<0$, while $n=0$ has the finite-root sign. Imaginary positive roots are exactly $n\delta$, $n\ge 1$, because $\delta ={\alpha }_0+\theta \in Q_{+}$; F3 gives their multiplicity $\mathrm{\ell }$. The value $n=0$ gives Cartan weight zero and is not a root.

Split the finite roots in step 1.1 into $\alpha \in {\mathrm{\Phi }}_0^{+}$ and their negatives. Their exponentials are respectively $e^{-\alpha }{q}^n$ for $n\ge 0$ and $e^{\alpha }{q}^n$ for $n\ge 1$. Imaginary roots contribute $q^n$ for $n\ge 1$. Inserting these disjoint exhaustive families, with their multiplicities, in F1's product gives the displayed expression. No root is lost or repeated, and regrouping is permitted because each coefficient has only finitely many contributing root factors, as in F1. Rank one gives exponent one on the imaginary product, whereas larger rank retains $\mathrm{\ell }$. This is a formal reindexing with no analytic convergence or AC assumption.