I am new to Lie algebra and have a maybe naive question about root system.
If we have a root system $\Phi$ we can associate it with a lattice $\Lambda(\Phi)$. I want to know how to recover the $\Phi$ from $\Lambda(\Phi)$? In particular I want to know that for type ADE, is $\Phi$ just the vectors in $\Lambda$ with minimal norm?
Edit
I note that my original question does not make much sense because non-isomorphic root systems may give the same lattice, for example $A_1\times A_1$ and $B_2$. So let's only consider the case of type ADE.
The vectors in $\Lambda$ of minimal norm ($>0$) are usually normalized to have norm $\sqrt{2}$ and give you back precisely the root system $\Phi$ as it is proved case by case for type $\mathsf{ADE}$ in the Wikipedia page on root systems.
It is not explicitly stated like this. But look under the section "Explicit construction of the irreducible root systems", read those subsection for type $\mathsf{ADE}$. Each time the root system is defined upon the root lattice by vectors of length $\sqrt{2}$.
(These are the comments of mine above made into an answer.)