Dynkin diagram with edge multiplicity and arrow convention

Let $\mathrm{\Phi }\subseteq E$ be a reduced crystallographic root system with base $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ and Cartan matrix $A=(a_{ij})$ (Properties of finite-type Cartan matrices). The Dynkin diagram of $\mathrm{\Phi }$ relative to $\mathrm{\Delta }$ is the graph with vertex set $\{1,\dots ,r\}$, with $a_{ij}{a}_{ji}$ edges joining the vertices $i$ and $j$ for $i\mathrm{\ne }j$, and with a decoration of the edges when $a_{ij}{a}_{ji}\mathrm{\ne }0$: if $a_{ij}{a}_{ji}=1$ no decoration is used; if $a_{ij}{a}_{ji}=2$ the two parallel edges carry a single arrow pointing from the longer root to the shorter root, that is, toward the vertex $j$ with $\mathrm{\mid }{\alpha }_j\mathrm{\mid }<\mathrm{\mid }{\alpha }_i\mathrm{\mid }$; if $a_{ij}{a}_{ji}=3$ the three parallel edges carry the same arrow toward the shorter root. The numbers $a_{ij}{a}_{ji}\in \{0,1,2,3\}$ are determined by the Cartan matrix and the arrow direction is determined by which of $a_{ij},a_{ji}$ is larger in absolute value, since $\mathrm{\mid }{\alpha }_i{\mathrm{\mid }}^2\mathrm{/}\mathrm{\mid }{\alpha }_j{\mathrm{\mid }}^2=a_{ji}\mathrm{/}{a}_{ij}$ whenever $a_{ij}{a}_{ji}\mathrm{\ne }0$ (Rank-two root-system classification); hence the diagram, with its multiplicities and arrows, is determined by $A$. Isolated vertices, that is, simple roots orthogonal to all others, are allowed and correspond to one-dimensional direct summands.

Conversely the Cartan matrix is recovered from the diagram: a pair with no edge has $a_{ij}=a_{ji}=0$; a pair joined by $m=a_{ij}{a}_{ji}$ edges has $a_{ij},a_{ji}<0$ with product $m$, and the arrow, which records which of $\mathrm{\mid }{a}_{ij}\mathrm{\mid },\mathrm{\mid }{a}_{ji}\mathrm{\mid }$ is larger, fixes $a_{ij}$ and $a_{ji}$ uniquely.