Shortest vectors in a root lattice

Let $R$ be a simply-laced root system in a Euclidean vector space $E$, with inner product normalized so that every root has length $\sqrt{2}$. Let $L \subseteq E$ be the lattice spanned by $R$. Is it true that $R$ is the root system of $L$ (i.e., that $R$ is the set of elements of $L$ of length $\sqrt{2}$)?