# Cayley plane

In mathematics, the **Cayley plane** (or **octonionic projective plane**) **P**^{2}(**O**) is a projective plane over the octonions.^{[1]} It was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley (for his 1845 paper describing the octonions).

More precisely, there are two objects called Cayley planes, namely the real and the complex Cayley plane.
The **real Cayley plane** is the symmetric space F_{4}/Spin(9), where F_{4} is a compact form of an exceptional Lie group and Spin(9) is the spin group of nine-dimensional Euclidean space (realized in F_{4}). It admits a cell decomposition into three cells, of dimensions 0, 8 and 16.^{[2]}

The **complex Cayley plane** is a homogeneous space under a noncompact (adjoint type) form of the group E_{6} by a parabolic subgroup *P*_{1}. It is the closed orbit in the projectivization of the minimal representation of E_{6}. The complex Cayley plane consists of two F_{4}-orbits: the closed orbit is a quotient of F_{4} by a parabolic subgroup, the open orbit is the real Cayley plane.^{[3]}

In the Cayley plane, lines and points may be defined in a natural way so that it becomes a 2-dimensional projective space, that is, a projective plane. It is a non-Desarguesian plane, where Desargues' theorem does not hold.