Recently, I am studying Lie algebra and have some question for the root system.
Let $\mathfrak g$ be a simple Lie algebra and $\mathfrak h$ a cartan subalgebra and $\Delta$ the root system of $\mathfrak g$
Let $O_{\mathfrak g}$ be the sublattice of the weight lattice spanned by long roots and $\{\alpha_1,\dots,\alpha_l\}$ a set of simple roots.
Weight lattice $\Lambda_w$ is the set $\{\lambda \in \mathfrak h^*|\langle\lambda, \alpha^\vee\rangle\in \mathbb Z \text{ for all } \alpha\in R\}$
($\alpha^{\vee}=\frac{2}{\langle\alpha,\alpha\rangle}\alpha$, and $\langle\text{ },\text{ }\rangle$ is an invariant bilinear form on $\mathfrak g$ normalized by $\langle \theta,\theta\rangle=2$, where $\theta$ is the highest root.)
In the irreducible root system, there are two lengths on roots. Since root system of a simple lie algebra is irreducible, we can think long roots and short roots.(In the semisimple case, we cannot say long and short root?)
Author says that
Lemma: The sublattice $O_\mathfrak g\subset\mathfrak h$ is generated by $\{\frac{2}{\langle\alpha,\alpha\rangle}\alpha|\alpha\in\Delta\}$. If $\mathfrak g$ is $A_n,D_n,E_n(resp. B_n,C_n,F_4,G_2)$, then $O_\mathfrak g$ is isomorphic to the root lattice(resp. $D_n, A_1^n,D_4,A_2).$
I understand the lemma as follows(Actually I don't know how to prove the lemma).
$O_\mathfrak g$ is a coroot lattice and in case of $\mathfrak g \text{ is }A_n,D_n,E_n$, this coroot lattice is isomorphic the the root lattice(Is it right?)
However $O_\mathfrak g$ is spanned by long roots and simple roots which are all element in $\Delta$ and this gives me confusion.
Since $O_\mathfrak g$ already contains all simple roots and all roots are $\mathbb{Z}$-linear combination of simple roots,
I think that $O_\mathfrak g$ is just a root lattice.
Does I misunderstand some basic concept?