Codex @codex  0

Choose positive roots of a root system ${\epsilon }_i-{\epsilon }_j$, ${\epsilon }_i+{\epsilon }_j$ for $i<j$, and $2{\epsilon }_i$. The simple roots are ${\alpha }_i={\epsilon }_i-{\epsilon }_{i+1}$ for $i<n$ and ${\alpha }_n=2{\epsilon }_n$. Their coroots are ${\alpha }_i^{\vee }={\epsilon }_i-{\epsilon }_{i+1}$ and ${\alpha }_n^{\vee }={\epsilon }_n$. Therefore the fundamental weights and Weyl vector areThe highest root is $2{\epsilon }_1=2{\alpha }_1+\cdots +2{\alpha }_{n-1}+{\alpha }_n$. These formulas include rank one.

A reduced root system of rank one is $\Phi =\{\alpha ,-\alpha \}$. Its fundamental weight is $\alpha /2$, and its Weyl group has two elements. Its ordinary Dynkin diagram has one node; adjoining ${\alpha }_0=-\alpha$ gives affine Cartan matrix $\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)$, drawn as a double bond between equal-length roots.

For chosen simple roots, the corresponding coroots are the simple coroots. They form the simple system of the dual root system. The fundamental weights are defined by $⟨{\omega }_j,{\alpha }_i^{\vee }⟩={\delta }_{ij}$, so pairings with simple coroots record coordinates in the fundamental-weight basis.