Remark

Every element of P defines a simple $\mathfrak{h}$-module of dimension 1, hence a homomorphism from the group $\mathbf{Z}[P]$ to the group $\mathscr{R}(\mathfrak{h})$, which is an injective homomorphism of rings. It is immediate that the composite

is the homomorphism defined by the canonical injection of $\mathfrak{h}$ into $\mathfrak{g}$ (no. 6).

The Weyl group W operates by automorphisms on the group P, and hence operates on $\mathbf{Z}^P$. For all $\lambda \in P$ and all $w\in W$, we have $we^{\lambda}=e^{w\lambda}$. Let $\mathbf{Z}[P]^W$ be the subring of $\mathbf{Z}[P]$ consisting of the elements invariant under W.