For $n$ an integer divisible by $8$, let me denote by $E_n$ the "usual" even non-degenerate positive definite integral symmetric bilinear form over $\mathbf Z^n$.

It is well known that in dimension 16, the only even non-degenerate positive definite integral symmetric bilinear form are (up to isometry) $E_8 \oplus E_8$ and $E_{16}$. Moreover, it is known that these two have same theta series (i.e. their quadratic forms represent the same integers, the same amount of times).

My question is thus: given a dimension 16 non-degenerate positive definite integral symmetric bilinear form, what is the easiest test to determine whether it is isometric to $E_8 \oplus E_8$ or to $E_{16}$?

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