  In this paper we construct 16 free algebras of modular forms on symmetric
domains of type IV for some reflection groups related to the eight lattices
$A_1(2)$, $A_1(3)$, $A_1(4)$, $2A_1(2)$, $A_2(2)$, $A_2(3)$, $A_3(2)$,
$D_4(2)$. As a corollary, we prove the modularity of formal Fourier--Jacobi
expansions for these reflection groups.
