I'm currently learning about gluing vectors in lattice theory, mainly from The (Sensual) Quadratic Form by Conway & Fung, and Sphere Packings, Lattices and Groups by Conway & Sloane. In the latter, in page 100, there is a theorem which states
Theorem 1: If a unimodular lattice $L$ is formed by gluing together two lattices $L_1$ and $L_2$ in such a way that there is no self-glue, i.e., if
$$L_1 = (L_1 \otimes \mathbb{R}) \cap L, ~~ L_2 = (L_2 \otimes \mathbb{R}) \cap L,$$
then the dual quotients $L_1^*/L_1$ and $L_2^*/L_2$ are isomorphic groups.
In working with the $E_8$ lattice, one finds rank 8 sublattices by using the extended diagram algorithm, getting the eight lattices
$$A_8,~A_1A_2A_5,~A_4^2,~D_5A_3,~E_6A_2,~E_7A_1,~A_1A_7,~D_8$$
Of these, four are products of two lattices and can be extended to $E_8$ again by including a glue vector corresponding to the simple root which was deleted in the extended diagram algorithm. Now, I find that Theorem 1 holds in the cases $$E_7A_1, ~~~~~~~~~~~~~~~ E_7^*/E_7 \simeq A_1^*/A_1 \simeq \mathbb Z_2,$$ $$E_6A_2, ~~~~~~~~~~~~~~~ E_6^*/E_6 \simeq A_2^*/A_2 \simeq \mathbb Z_3,$$ $$D_5A_3, ~~~~~~~~~~~~~~~ D_5^*/D_5 \simeq A_3^*/A_3 \simeq \mathbb Z_4.$$
However, for the case $A_1A_7$ the dual quotients are $\mathbb Z_2$ and $\mathbb Z_8$, which are obviously not isomorphic. Given that the glue vector required for extending $A_1A_7 \to E_8$ is not orthogonal to either of these sublattices, Theorem 1 should hold.
Can anyone please tell me what am I doing wrong here?
Incidentally, the lattices $A_1A_2$ and $A_5$ have isomorphic dual quotients $\mathbb Z_2 \times \mathbb Z_3$ and $\mathbb Z_6$. Does this mean anything?
The reason that Theorem 1 does not hold for the sublattice $A_1\oplus A_7$ is that $A_7$ does have a self-glue in $E_8$.
Let us define the $E_8$ lattice as the set of vectors $v = (v_1,...,v_8)$ such that all $v_i$'s are simultaneously integer or half-integer, and $\sum_{i = 1}^8 v_i \in 2 \mathbb Z$, as usual.
An embedding of $A_1\oplus A_7$ into $E_8$ is given by the basis vectors
$$ A_1: (\tfrac12, \tfrac12, \tfrac12, \tfrac12, \tfrac12, \tfrac12, \tfrac12, \tfrac12) $$
and
$$A_7: (1,-1,0,...,0), (0,1,-1,0,...,0),...,(0,...,0,1,-1). $$
Now, the real span of this particular $A_7$ is the set of points $\{x = (x_1,...,x_8) | \sum_{i = 1}^8 x_i = 0\}$, which contains in particular the vector $v = (\tfrac12, \tfrac12, \tfrac12, \tfrac12, -\tfrac12, -\tfrac12, -\tfrac12, -\tfrac12)$. This vector is a self-glue for $A_7$, extending it to $E_7$, and we recover the case $A_1 \oplus E_7 \subset E_8$, which satisfies Theorem 1.
In the theory of lattices we would equivalently say that $A_7$ is not primitively embedded into $E_8$.
The lattices $A_1 \oplus A_2$ and $A_5$ can be shown to be primitively embedded in $E_8$ such that they are mutually orthogonal, hence they satisfy Theorem 1.