Let $\mathfrak g$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field $k$ of characteristic zero, let $\mathfrak h \subset \mathfrak g$ be a Cartan subalgebra, and let

\begin{align*} \Phi := \{\alpha \in \mathfrak h^* \setminus \{0\} : \mathfrak g_\alpha \neq 0\} \end{align*}

be the set of non-zero roots for the root space decomposition of $\mathfrak g$ relative to $\mathfrak h$. Let $E := \operatorname{span}_{\mathbb R}\Phi$ be the real [vector space](/page/Vector%20Space) spanned by the roots, equipped with the positive definite symmetric [bilinear form](/page/Bilinear%20Form) $(\cdot,\cdot)$ induced by the Killing form through the standard real form of the root lattice. Then $\Phi \subset E$ is a reduced crystallographic root system: $\Phi$ is finite, spans $E$, contains no zero vector, is invariant under the root reflections

\begin{align*} s_\alpha: E &\to E,\\ \beta &\mapsto \beta - \frac{2(\beta,\alpha)}{(\alpha,\alpha)}\alpha \end{align*}

for every $\alpha \in \Phi$, satisfies

\begin{align*} \frac{2(\beta,\alpha)}{(\alpha,\alpha)} \in \mathbb Z \end{align*}

for all $\alpha,\beta \in \Phi$, and has the property that if $\alpha \in \Phi$ and $c\alpha \in \Phi$ for some $c \in \mathbb R$, then $c = \pm 1$.