I have a rather stupid lattice theory question. Suppose $L$ is a root lattice that can be primitively embedded in the $E_8$ lattice. Is the orthogonal complement of $L$ in $E_8$ unique up to isomorphism, or for different primitive embeddings could I get non-isomorphic complements?
I have a rather stupid lattice theory question. Suppose $L$ is a root lattice that can be embedded in the $E_8$ lattice. Is the orthogonal complement of $L$ in $E_8$ unique up to isomorphism, or for different embeddings could I get non-isomorphic complements?