Proposition 26

The set of $x \in V$ such that $\langle x, y^* \rangle \in \mathbf{Z}$ for all $y^* \in Q(\mathbf{R}^*)$ (or, equivalently, for all $y^* \in \mathbf{R}^*$) is a discrete subgroup $G$ of $V$ containing $Q(\mathbf{R})$. If $B'$ is a basis of $\mathbf{R}^*$, the basis of $V$ dual to $B'$ is a basis of $G$.

Let $x \in V$. The following three properties are equivalent:

(i) $\langle x, y^* \rangle \in \mathbf{Z}$ for all $y^* \in Q(\mathbf{R}^*)$; (ii) $\langle x, y^* \rangle \in \mathbf{Z}$ for all $y^* \in B'$; (iii) the coordinates of $x$ with respect to the dual basis of $B'$ are in $\mathbf{Z}$.

We deduce from this that the basis dual to $B'$ is a basis of $G$. On the other hand, (RSIII) proves that $\mathbf{R} \subseteq G$, so $Q(\mathbf{R}) \subseteq G$.

The group $G$ of Prop. 26 is denoted by $P(\mathbf{R})$, and its elements are called the weights of $\mathbf{R}$. We can also consider the group $P(\mathbf{R}^*)$ of weights of $\mathbf{R}^*$.

are finite groups in duality over $\mathbf{Q}/\mathbf{Z}$, and hence are isomorphic. The common order of these two groups is called the connection index of R (or of $\tilde{R}$).

If R is a direct sum of root systems $R_i$, the group Q(R) (resp. P(R)) is identified canonically with the direct sum of the Q($R_i$) (resp. P($R_i$)).