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 nonisomorphic complements?
You can get different orthogonal complements for different embeddings. There are two different embeddings of $A_{7}$ in $E_{8}$ so that for the first embedding the orthogonal complement is the lattice $A_{1}$, and for the second embedding the orthogonal complement is the lattice $\langle 8 \rangle$. 

