Let $\mathfrak g$ be a finite-dimensional complex semisimple Lie algebra, let $\mathfrak h \subset \mathfrak g$ be a Cartan subalgebra, and let $\Phi \subset \mathfrak h^*$ be the corresponding root system. For $\alpha \in \Phi$, let $\alpha^\vee \in \mathfrak h$ denote the coroot, so that $\alpha(\alpha^\vee)=2$. Let $\alpha,\beta \in \Phi$ satisfy $\beta \ne \alpha$ and $\beta \ne -\alpha$.

Suppose that the $\alpha$-string through $\beta$ consists exactly of the roots

\begin{align*} \beta - p\alpha,\ \beta-(p-1)\alpha,\ \ldots,\ \beta,\ \ldots,\ \beta+(q-1)\alpha,\ \beta+q\alpha, \end{align*}

where $p,q \in \mathbb N \cup \{0\}$, meaning that $\beta+n\alpha \in \Phi$ if and only if $-p \le n \le q$. Then

\begin{align*} \langle \beta,\alpha^\vee\rangle = p-q. \end{align*}