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In mathematics, the **Cayley plane** (or **octonionic projective plane**) **P**^{2}(**O**) is a projective plane over the octonions.^{[1]}

The Cayley plane was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley for his 1845 paper describing the octonions.

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.

More precisely, as of 2005, 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 the complexification of the group E_{6} by a parabolic subgroup *P*_{1}. It is the closed orbit in the projectivization of the minimal complex representation of E_{6}. The complex Cayley plane consists of two complex F_{4}-orbits: the closed orbit is a quotient of the complexified F_{4} by a parabolic subgroup, the open orbit is the complexification of the real Cayley plane,^{[3]} retracting to it.

- Baez, John C. (2002). "The Octonions".
*Bulletin of the American Mathematical Society*.**39**(2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087. - Iliev, A.; Manivel, L. (2005). "The Chow ring of the Cayley plane".
*Compositio Mathematica*.**141**: 146. arXiv:math/0306329. doi:10.1112/S0010437X04000788. - Ahiezer, D. (1983). "Equivariant completions of homogenous algebraic varieties by homogenous divisors".
*Annals of Global Analysis and Geometry*.**1**: 49–78. doi:10.1007/BF02329739. - Baez, John C. (2005). "Errata for
*The Octonions*" (PDF).*Bulletin of the American Mathematical Society*.**42**(2): 213–213. doi:10.1090/S0273-0979-05-01052-9. - McTague, Carl (2014). "The Cayley plane and string bordism".
*Geometry & Topology*.**18**(4): 2045–2078. arXiv:1111.4520. doi:10.2140/gt.2014.18.2045. MR 3268773. Zbl 1323.55007. - Helmut Salzmann et al. "Compact projective planes. With an introduction to octonion geometry"; de Gruyter Expositions in Mathematics, 21. Walter de Gruyter & Co., Berlin, 1995. xiv+688 pp. ISBN 3-11-011480-1