Cartan matrix of a based root system

Let $\mathrm{\Phi }\subseteq E$ be a reduced crystallographic root system with base $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ (Positive systems and simple roots) and coroots ${\alpha }_i^{\vee }=2{\alpha }_i\mathrm{/}({\alpha }_i,{\alpha }_i)$ (Coroot and dual root system). The Cartan matrix of $\mathrm{\Phi }$ relative to $\mathrm{\Delta }$ is the $r\times r$ matrix $A=(a_{ij})$ with rows indexed by coroots, $a_{ij}=({\alpha }_j,{\alpha }_i^{\vee })=\frac{2({\alpha }_j,{\alpha }_i)}{({\alpha }_i,{\alpha }_i)}\mathrm{.}$ Thus $a_{ij}$ is the Cartan integer of the ordered pair $({\alpha }_j,{\alpha }_i)$, that is, the coefficient of ${\alpha }_i$ subtracted from ${\alpha }_j$ in the reflection $s_{{\alpha }_i}({\alpha }_j)={\alpha }_j-a_{ij}{\alpha }_i$; it is not in general an eigenvalue of ${\alpha }_j$. All entries are integers by the root-system axioms, $a_{ii}=2$ for every $i$, and $a_{ij}\le 0$ for $i\mathrm{\ne }j$ (Distinct simple roots have nonpositive inner product). The Cartan matrix depends on the numbering of the simple roots: renumbering conjugates it by the corresponding permutation matrix. The indexing convention here is the row-coroot convention: the entries of row $i$ record the action of the coroot ${\alpha }_i^{\vee }$ on the other simple roots.