Corollary

The homomorphism $\varphi$ defines by passage to the quotient an isomorphism from the group $A(R)/W(R)$ to the group of automorphisms of the Dynkin graph of $R$.

Clearly, $\varphi(g) = \mathrm{Id}$ for all $g \in W(R)$. On the other hand, Prop. 1 shows that there exists an isomorphism $\psi$ from the group of isomorphisms of the Dynkin graph of $R$ to the subgroup $E$ of elements of $A(R)$ leaving fixed a given basis $B$ of $R$, such that $\varphi \circ \psi = \mathrm{Id}$. Since $A(R)$ is the semi-direct product of $E$ and $W(R)$ (§ 1, no. 5, Prop. 16), the corollary follows.

In practice, the Dynkin graph $(X, f)$ is represented by a diagram composed of nodes and bonds in the following way. The nodes correspond to the vertices of $X$; two nodes corresponding to two distinct vertices $i$ and $j$ are joined by 0, 1, 2 or 3 bonds in cases 1), 2), 3) and 4) above (up to interchanging $i$ and $j$). Moreover, in cases 3) and 4), that is when $f(i, j) > 1$, or when the roots $\alpha_i$ and $\alpha_j$ are not orthogonal and not of the same length, an inequality sign $>$ is placed on the double or triple bond joining the nodes corresponding to $i$ and $j$ oriented towards the node corresponding to $j$ (that is, the shortest root):

It is clear that the Dynkin graph $(X, f)$ can be recovered from this diagram.

We remark that the diagram associated to the Coxeter graph of $W(R)$ can be obtained from that associated to the Dynkin graph of $R$ by keeping the nodes and the single bonds and replacing the double (resp. triple) bonds by a bond surmounted by the number 4 (resp. 6). Conversely, given the Coxeter graph of W(R), the diagram associated to the Dynkin graph of R can be recovered using the inverse of this procedure, except for the inequality signs on the double or triple bonds. Th. 1 thus gives immediately the list of possible Dynkin graphs. More precisely: