Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$. Let for some fixed cartan subalgebra $\mathfrak{h}$, $\Phi$ be the root system of $\mathfrak{g}$. Let $\Delta=\{\alpha_1,\dots,\alpha_{\ell}\}$ be a simple system for $\Phi$. Let $P=\{\lambda\in\mathfrak{h}^*\mid (\lambda,\alpha^\vee)\in\mathbb{Z}\quad \forall \alpha\in\Delta\}$ be the weight lattice and $Q=Span_{\mathbb{Z}}\Delta$ be the root lattice. Then $Q\subseteq P$. If $w_1,w_2,\dots,w_{\ell}$ are the fundamental dominant weights, then $P$ is a lattice with basis $w_1,w_2,\dots,w_{\ell}$. Let $A$ be the Cartan matrix of $\Phi$.
My Question is: How to prove that the order of the group $P/Q$ is same as $det(A)$?
My progress so far :
I could express $\alpha_i$'s in terms of $w_i$s and coefficient are Cartan integers. By inverting the system I showed that $det(A)w_i\in Q$ for all $i$. But don't know how to prove that the order is $det(A)$.
For classical cases we have a very explicit description of $w_i$ s in terms of $\alpha_i$ s, there it's not hard to see that order is indeed $det(A)$. But I'm trying to find a case free solution.
For general case I tried to use the stacked basis theorem for finitely generated free abelian groups but could not come up with and ans.
Any suggestion is welcomed and appreciated. Thank you in advance.