Let $V$ be a finite-dimensional real [inner product space](/page/Inner%20Product%20Space) with inner product $(\cdot,\cdot)$, let $\Phi \subset V$ be a crystallographic root system, and let $\Delta=\{\alpha_1,\dots,\alpha_n\}$ be a simple root basis of $\Phi$. Define the Cartan matrix $A=(a_{ij}) \in \mathbb{Z}^{n \times n}$ by

\begin{align*} a_{ij} := \frac{2(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)} \end{align*}

for $1 \leq i,j \leq n$. Then:

1. $a_{ii}=2$ for every $1 \leq i \leq n$. 2. $a_{ij}\in \mathbb{Z}_{\leq 0}$ whenever $i\ne j$. 3. $a_{ij}=0$ if and only if $a_{ji}=0$. 4. There exist positive [real numbers](/page/Real%20Numbers) $d_1,\dots,d_n$ such that, for $D=\operatorname{diag}(d_1,\dots,d_n)$, the matrix $DA$ is symmetric.