From wikipedia: $\mathbb{CP}^n$ is a Symmetric space of type AIII for $p=n$, $q=1$. There are embeddings of both $U(1)$ and $SU(n)$ into $S(U(n) \times U(1)) \subset SU(n+1)$ which give you your quotient by right multiplication.

From wikipedia: $\mathbb{CP}^n$ is a Symmetric space of type AIII for $p=n$, $q=1$. There are embeddings of both $U(1)$ and $SU(n)$ into $S(U(n) \times U(1)) \subset SU(n+1)$ which give you your quotient by right multiplication.