Let $E$ be a finite-dimensional real [inner product space](/page/Inner%20Product%20Space), let $\Phi \subset E$ be a finite root system, and let $\Phi^+ \subset \Phi$ be a positive system, so that $\Phi = \Phi^+ \sqcup (-\Phi^+)$. Let
paragraph
admin
\begin{align*}
\Delta := \{\alpha \in \Phi^+ : \alpha \neq \beta + \gamma \text{ for all } \beta,\gamma \in \Phi^+\}
\end{align*}
latex_env
admin
be the set of simple roots in $\Phi^+$. Then $\Delta$ is a base of $\Phi$: the set $\Delta$ is linearly independent over $\mathbb{R}$, and every root $\rho \in \Phi$ can be written as