Kac moody integral and dominant integral weights

Fix a minimal realization $(\mathfrak{h},(h_i),({\alpha }_i))$ of a finite GCM $A$, with the row-coroot convention ${\alpha }_j(h_i)=a_{ij}$ from Realization of a generalized cartan matrix. Define

$P=\{\lambda \in {\mathfrak{h}}^{\ast }:\lambda (h_i)\in \mathbb{Z}\text{ for every }i\},P^{+}=\{\lambda \in P:\lambda (h_i)\ge 0\text{ for every }i\}\mathrm{.}$

Elements of $P$ are integral weights; elements of $P^{+}$ are dominant integral weights. Their simple-coroot labels are the integers $\lambda (h_i)$. Zero labels are allowed, and the zero functional is dominant integral.

The independent coroots can be extended to a finite basis of $\mathfrak{h}$. Prescribing integral labels on the coroots and arbitrary complex values on the remaining basis vectors specifies a unique functional. Thus complementary Cartan values are unrestricted; $P$ need not be a discrete lattice in ${\mathfrak{h}}^{\ast }$. The inequalities only concern the integral coroot labels, not an ordering of arbitrary complex Cartan values. This definition uses finite linear algebra and no AC.