Classical root systems in coordinates

Each set is finite, omits $0$, and is reduced. The differences ${\epsilon }_i-{\epsilon }_n$ span the sum-zero hyperplane for $A_{n-1}$; $B_n$ and $C_n$ contain a nonzero multiple of every coordinate vector; and in $D_n$, $({\epsilon }_i+{\epsilon }_j)+({\epsilon }_i-{\epsilon }_j)=2{\epsilon }_i$ for any $j\mathrm{\ne }i$, which exists because $n\ge 2$. Thus each set spans its stated Euclidean space.

Reflection closure: $s_{{\epsilon }_i}$ and $s_{2{\epsilon }_i}$ negate the $i$th coordinate and preserve $B_n,C_n,D_n$; $s_{{\epsilon }_i-{\epsilon }_j}$ swaps coordinates $i,j$ and preserves all four sets; and $s_{{\epsilon }_i+{\epsilon }_j}$ swaps and negates those two coordinates and preserves $B_n,C_n,D_n$. These are precisely the root reflections that occur in the displayed sets.

Integrality: proportional pairs give Cartan integer $\pm 2$. For nonproportional pairs, roots of squared length $2$ pair by $0$ or $\pm 1$; a short root of squared length $1$ in $B_n$ pairs by $0$ or $\pm 1$; and a long root $2{\epsilon }_i$ of squared length $4$ in $C_n$ pairs with a mixed root by $0$ or $\pm 2$. Hence every Cartan integer is in $\{0,\pm 1,\pm 2\}$. Together with steps 1.1 and 1.2, [L1] proves that all four displayed sets are reduced crystallographic root systems.

The listed simple roots of [L3] have the standard Cartan matrices of $A_{n-1},B_n,C_n,D_n$ (with $D_2=A_1\bigsqcup A_1$), giving the stated Dynkin diagrams; and [L2] identifies these coordinate sets as the root systems of ${\mathfrak{s}\mathfrak{l}}_n,{\mathfrak{s}\mathfrak{o}}_{2n+1},{\mathfrak{s}\mathfrak{p}}_{2n},{\mathfrak{s}\mathfrak{o}}_{2n}$ respectively.