Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and root system $\Phi \subset \mathfrak{h}^*$. For each $\alpha \in \Phi$, let $h_\alpha \in \mathfrak{h}$ be the coroot element belonging to an $\mathfrak{sl}_2$-triple $(e_\alpha,h_\alpha,f_\alpha)$ with $e_\alpha \in \mathfrak{g}_\alpha$ and $f_\alpha \in \mathfrak{g}_{-\alpha}$. Let $\alpha,\beta \in \Phi$ satisfy $\beta \ne \alpha$ and $\beta \ne -\alpha$. Let $r,q \in \mathbb{Z}_{\ge 0}$ be the maximal integers such that

\begin{align*} \beta-r\alpha,\ldots,\beta,\ldots,\beta+q\alpha \end{align*}

are precisely the roots in $\Phi$ of the form $\beta+n\alpha$ with $n \in \mathbb{Z}$. Then

\begin{align*} \beta(h_\alpha)=r-q. \end{align*}

In particular, $\beta(h_\alpha)\in\mathbb{Z}$.