Positive definite lattices of rank at most 8

We give a short uniqueness proof for the E8 root lattice, and in fact for all positive definite unimodular lattices of rank up to 8. Our proof is done with elementary arguments, mainly these: (1) invariant theory for integer matrices; (2) an upper bound for the minimum of nonzero norms (either of the elementary bounds of Hermite or Minkowski will do). We make no use of p-adic completions, mass formulas or modular forms.