For any compact simple Lie groupG, there is a unique G/H obtained as a quotient of G by a subgroup
Here, Sp(1) is the compact form of the SL(2)-triple associated with the highest root of G, and K its centralizer in G. These are classified as follows.
G
H
quaternionic dimension
geometric interpretation
p
Grassmannian of complex 2-dimensional subspaces of
p
Grassmannian of oriented real 4-dimensional subspaces of
p
Grassmannian of quaternionic 1-dimensional subspaces of
These spaces can be obtained by taking a projectivization of
a minimal nilpotent orbit of the respective complex Lie group.
The holomorphic contact structure is apparent, because
the nilpotent orbits of semisimple Lie groups
are equipped with the Kirillov-Kostant holomorphic symplectic form. This argument also explains how one
can associate a unique Wolf space to each of the simple
complex Lie groups.
Besse, Arthur L. (2008), Einstein Manifolds, Classics in Mathematics, Berlin: Springer-Verlag, ISBN978-3-540-74120-6, MR2371700. Reprint of the 1987 edition.