  We prove that the quantum group inclusion $O_n \subset O_n^*$ is "maximal",
where $O_n$ is the usual orthogonal group and $O_n^*$ is the half-liberated
orthogonal quantum group, in the sense that there is no intermediate compact
quantum group $O_n\subset G\subset O_n^*$. In order to prove this result, we
use: (1) the isomorphism of projective versions $PO_n^*\simeq PU_n$, (2) some
maximality results for classical groups, obtained by using Lie algebras and
some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.
