Let $E_6$ be the root lattice, and $G$ be its automorphism group as a lattice (i.e. as $\mathbb Z$-module together with the inner product). Let $W(E_6)$ be the Weyl group. Apparently $W(E_6)\subset G$. Is it in fact an equality?
I know in general we cannot expect this for every root lattice, but is it true for this case? Thanks in advance.