We prove a necessary and sufficient condition for the graded algebra of automorphic forms on a symmetric domain of type IV to be free. From the necessary condition, we derive a classification result. Let M be an even lattice of signature (2, n) splitting two hyperbolic planes. Suppose Γ is a subgroup of the integral orthogonal group of M containing the discriminant kernel. It is proved that there are exactly 26 groups Γ such that the space of modular forms for Γ is a free algebra. Using the sufficient condition, we recover some well-known results.

For 25 orthogonal groups of signature (2, n) related to the root lattices A1,

We give an explicit formula to express the weight of 2-reflective modular forms. We prove that there is no 2-reflective lattice of signature (2, n) when n ≥ 15 and n = 19 except the even unimodular lattices of signature (2, 18) and (2, 26). As applications, we give a simple proof of Looijenga's theorem that the lattice 2U ⊕ 2E 8 (−1) ⊕ −2n is not 2-reflective if n > 1. We also classify reflective modular forms on lattices of large rank and the modular forms with the simplest reflective divisors.

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