  We study the algebras of modular forms on type IV symmetric domains for
simple lattices; that is, lattices for which every Heegner divisor occurs as
the divisor of a Borcherds product. For every simple lattice $L$ of signature
$(n,2)$ with $3\leq n \leq 10$, we prove that the graded algebra of modular
forms for the maximal reflection subgroup of the orthogonal group of $L$ is
freely generated. We also show that, with five exceptions, the graded algebra
of modular forms for the maximal reflection subgroup of the discriminant kernel
of $L$ is also freely generated.
