  We construct $16$ reflection groups $\Gamma$ acting on symmetric domains
$\mathcal{D}$ of Cartan type IV, for which the graded algebras of modular forms
are freely generated by forms of the same weight, and in particular the
Satake--Baily--Borel compactification of $\mathcal{D} / \Gamma$ is isomorphic
to a projective space. Four of these are previously known results of
Freitag--Salvati Manni, Matsumoto, Perna and Runge. In addition we find several
new modular groups of orthogonal type whose algebras of modular forms are
freely generated.
