Unimodularization of lattice by Kneser

Question

Let $L$ be a lattice in $\mathbb{R}^{n}$ corresponding to some root system (or Lie algebra, or Dyinkin diagram, etc...) I just learned that we can unimodularize $L$ by adding some shifted points. For example, the lattice corresponds to $A_{1}\oplus A_{1}$ is a lattice in $\mathbb{R}^{2}$ generated by $v_{1}=(\sqrt{2}, 0)$ and $v_{2}=(0, \sqrt{2})$. Then this Lattice can be unimodularized by considering $$ L^{+} := L \cup \left(L + \frac{1}{2}(v_{1}+v_{2})\right). $$ Then what is the corresponding process in the Lie algebra side (or root system or Dynkin diagram)? Thanks in advance.