Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 第 149 页 · Chapter II，Section 5「Abstract Root Systems」（本书的定义允许非约化根系）
An abstract root system in a finite-dimensional real inner product space $V$ with inner product $\langle\,\cdot\,,\,\cdot\,\rangle$ and norm squared $|\,\cdot\,|^2$ is a finite set $\Delta$ of nonzero elements of $V$ such that (i) $\Delta$ spans $V$, (ii) the orthogonal transformations $s_\alpha(\varphi)=\varphi-\frac{2\langle\varphi,\alpha\rangle}{|\alpha|^2}\alpha$, for $\alpha\in\Delta$, carry $\Delta$ to itself, (iii) $\frac{2\langle\beta,\alpha\rangle}{|\alpha|^2}$ is an integer whenever $\alpha$ and $\beta$ are in $\Delta$.