The BGG Euler identity gives the Weyl numerator

Given: The Axiom of Choice, a dominant integral weight $\lambda \in {\mathrm{\Lambda }}^{+}$, the module $L(\lambda )$ with its formal character, the Weyl vector $\rho$, the positive system ${\mathrm{\Phi }}^{+}$, and the alternants $A(\nu )$.

The Axiom of Choice is assumed; it enters through the BGG Euler identity of [F1] (The Axiom of Choice).

The BGG Euler identity in character form reads $\mathrm{ch}L(\lambda )={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}{e}^{w\circ \lambda }{\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha }{)}^{-1}$ in $\mathcal{R}$, where $w\circ \lambda =w(\lambda +\rho )-\rho$ (The Euler-character identity for a finite-dimensional simple module, The Grothendieck group and character of O).

The denominator identity gives $A(\rho )=e^{\rho }{\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha })$ and $A(\rho {)}^{-1}=e^{-\rho }{\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha }{)}^{-1}$ (The Weyl denominator identity), and the product ${\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha })$ is invertible (Geometric series are invertible in the completed character ring).

$A(\lambda +\rho )={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}{e}^{w(\lambda +\rho )}$ is a finite sum, $\mathrm{ch}L(\lambda )$ is a finite-support element, and $w\circ \lambda +\rho =w(\lambda +\rho )$ (The Weyl alternation operator, The formal character of a finite-dimensional weight module).

For $\lambda \in {\mathrm{\Lambda }}^{+}$ one has $\rho \in P$ and $\lambda +\rho \in P$, and $W$ preserves $P$, so all exponents of the two sides lie in the weight lattice and the identity is an identity of finite sums in $\mathbb{Z}[P]$ (Integral, dominant, and strictly dominant weights, Finite Weyl positive roots and simple reflections, The Weyl vector rho for a chosen positive system).