Root, coroot, weight, and coweight lattices

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, Simple roots form a signed integral basis) and coroots ${\alpha }^{\vee }=2\alpha \mathrm{/}(\alpha ,\alpha )\in E$ (Coroot and dual root system). The root lattice, coroot lattice, weight lattice, and coweight lattice are $Q={\sum }_{\alpha \in \mathrm{\Phi }}\mathbb{Z}\alpha ={\sum }_{i=1}^r\mathbb{Z}{\alpha }_i,Q^{\vee }={\sum }_{\alpha \in \mathrm{\Phi }}\mathbb{Z}{\alpha }^{\vee }={\sum }_{i=1}^r\mathbb{Z}{\alpha }_i^{\vee },$ $P=\{\lambda \in E:(\lambda ,{\alpha }^{\vee })\in \mathbb{Z}\text{ for all }\alpha \in \mathrm{\Phi }\},P^{\vee }=\{\mu \in E:(\alpha ,\mu )\in \mathbb{Z}\text{ for all }\alpha \in \mathrm{\Phi }\}\mathrm{.}$ All four sets are additive subgroups of $E$, since the integrality conditions are preserved by addition and negation; they are free abelian groups of rank $r$, for $Q$ and $Q^{\vee }$ because the simple roots and the simple coroots are bases of $E$, and for $P$ and $P^{\vee }$ because the simple coroots are again a basis so the dual lattice is generated by the dual basis. The chain $Q\subseteq P,Q^{\vee }\subseteq P^{\vee }$ holds because the Cartan integers $⟨\alpha ,{\beta }^{\vee }⟩=2(\alpha ,\beta )\mathrm{/}(\beta ,\beta )$ are integers for all roots (Reduced crystallographic Euclidean root system).