  A modular form on an even lattice $M$ of signature $(l,2)$ is called
reflective if it vanishes only on quadratic divisors orthogonal to roots of
$M$. In this paper we show that every reflective modular form on a lattice of
type $2U\oplus L$ induces a root system satisfying certain constrains. As
applications, (1) we prove that there is no lattice of signature $(21,2)$ with
a reflective modular form and that $2U\oplus D_{20}$ is the unique lattice of
signature $(22,2)$ and type $U\oplus K$ which has a reflective Borcherds
product; (2) we give an automorphic proof of Shvartsman and Vinberg's theorem,
asserting that the algebra of modular forms for an arithmetic subgroup of
$\mathrm{O}(l,2)$ is never freely generated when $l\geq 11$. We also prove
several results on the finiteness of lattices with reflective modular forms.
