It is a non-Desarguesian plane, where Desargues' theorem does not hold.
The real Cayley plane is the symmetric space F4/Spin(9), where F4 is a compact form of an exceptional Lie group and Spin(9) is the spin group of nine-dimensional Euclidean space (realized in F4).
[2] The complex Cayley plane is a homogeneous space under the complexification of the group E6 by a parabolic subgroup P1.
It is the closed orbit in the projectivization of the minimal complex representation of E6.
The complex Cayley plane consists of two complex F4-orbits: the closed orbit is a quotient of the complexified F4 by a parabolic subgroup, the open orbit is the complexification of the real Cayley plane,[3] retracting to it.