Indefinite lattices
Isotropic, hyperbolic and reflective lattices.
Let \(L\) be an indefinite lattice.
Isotropic lattices
\(L\) is isotropic when \(b(x, x) = 0\) for some nonzero \(x \in L\), and anisotropic otherwise.
Hyperbolic lattices
A lattice of rank \(n + 1 \geq 2\) is hyperbolic when its form is nondegenerate with signature \((1, n)\) or \((n, 1)\).
Reflective lattices
A hyperbolic lattice \(L\) is reflective when the subgroup of \(O(L)\) generated by the reflections \(s_r\), \(r \in \Phi(L)\), has finite index in \(O(L)\).
This property is declared; the notes cite its source.