 Orthogonal complex line flag manifold

For $n\ge 2$, ordered orthogonal complex lines in ${\mathbb{C}}^n$ form a projective bundle over ${\mathbb{CP}}^{n-1}$, with fiber ${\mathbb{CP}}^{n-2}$. Here $L$ is the first tautological line and the complement is formed using the standard Hermitian inner product. Its real dimension is $4n-6$. It is also the partial complex flag space of a line contained in a two-plane: send an orthogonal pair to $({\ell }_1,{\ell }_1\oplus {\ell }_2)$, and recover the second line by orthogonal complement inside the two-plane.