Classification of indefinite forms

Without proof, we state the result:

Theorem. Indefinite quadratic forms are completely determined by their type, rank and signature.

Odd forms thus are of the form $$Q=b_+\cdot\langle 1\rangle \perp b_-\cdot\langle -1\rangle$$ of rank $r=b_++b_-$ and signature $\sigma=b_+-b_-$.

Even forms are of the form $$Q=a\cdot \left(\pm E_8\right) \perp l\cdot H$$ of rank $r=8a+2l$ and signature $\sigma=\pm 8a$.

Note that

  • $E_8 \perp (-E_8) = 8H$
  • $\sigma(Q) \equiv 0 \mbox{ mod } 8$ if $Q$ is even.