 Space of orthogonal complex structures

On a Euclidean real space of dimension $2n$, an orthogonal complex structure is an orthogonal endomorphism $J$ with $J^2=-I$. Orthogonal changes of basis act transitively on these structures, and the stabilizer of the standard structure is $U(n)$. The compact homogeneous space has real dimension $n(n-1)$.

// Target: geometry-and-topology.bigb