Octonionic Projective Plane

• Octonionic projective plane is a 16-dimensional Riemannian symmetric space coordinatized by non-associative octonions, central to exceptional geometry.
• It is constructed via one-dimensional octonionic submodules and primitive idempotents in the Albert algebra, exhibiting a unique Moufang incidence structure.
• Its symmetric space realization as F4/Spin(9) links deep topological, combinatorial, and Lie group properties with connections to the Freudenthal–Tits magic square.

The octonionic projective plane, denoted $\mathbb{O}P^2$ and also known as the Cayley plane, is a 16-dimensional Riemannian symmetric space that serves as the canonical example of a projective plane coordinatized by the non-associative, alternative division algebra of octonions. It is the unique compact projective Moufang plane not coordinatizable by a field or associative division algebra. As the "top" of the family $\mathbb{K} P^2$ for normed division algebras $$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$, $\mathbb{O}P^2$ is central in exceptional geometry, the theory of exceptional Lie groups, and in the topology of high-dimensional manifolds.

1. Algebraic Construction and Coordinate Models

A point of $\mathbb{O}P^2$ is defined as a one-dimensional right $\mathbb{O}$-submodule of $\mathbb{O}^3$; equivalently, as the equivalence class $[x_1:x_2:x_3]$ of nonzero triples under right multiplication by invertible octonions: $$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$ Because the octonions $\mathbb{O}$ are non-associative but alternative, this quotient is well defined, and every two elements generate an associative subalgebra. Local affine charts are constructed as in the associative cases: on $\mathbb{K} P^2$0,

$\mathbb{K} P^2$1

so each chart is diffeomorphic to $\mathbb{K} P^2$2. Transition functions reduce to rational expressions using only two arguments at a time, exploiting alternativity and norm multiplicativity (Lackmann, 2019, Corradetti et al., 2022, Corradetti et al., 2023).

In the algebraic-geometric formulation, a point of $\mathbb{K} P^2$3 corresponds bijectively to a rank-one primitive idempotent $\mathbb{K} P^2$4 in the 27-dimensional exceptional Jordan (Albert) algebra $\mathbb{K} P^2$5 of $\mathbb{K} P^2$6 Hermitian octonionic matrices. Explicitly, $\mathbb{K} P^2$7, $\mathbb{K} P^2$8, and $\mathbb{K} P^2$9 (Corradetti et al., 2022, Corradetti et al., 2023, Chester et al., 1 Dec 2025). Homogeneous coordinates correspond to normalized outer products: $$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$0 with $$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$1, $$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$2.

2. Incidence Geometry and Moufang Property

Lines in $$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$3 are dual to its points. In the Veronese model, a point is specified by a Veronese vector $$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$4 satisfying nine homogeneous quadratic relations: $$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$5 (Corradetti et al., 2023, Corradetti et al., 2022). A line is the orthogonal complement to a Veronese vector under the form

$$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$6

Every two points lie on a unique line and, dually, every two lines meet in a unique point. The automorphism group acts transitively on flags (incident point-line pairs). The Moufang property holds: the group generated by elations with a given axis acts transitively on the flags with this axis, and all classical projective axioms are satisfied, due to octonionic alternativity (Corradetti et al., 2023, Lackmann, 2019).

3. Symmetric Space Structure and Isometry Groups

$$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$7 is realized as the Riemannian symmetric space

$$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$8

where $$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$$9 is the compact real form of the exceptional simple Lie group of type $\mathbb{O}P^2$0 (dimension 52), and $\mathbb{O}P^2$1 is the stabilizer of a base point (dimension 36). The real dimension is thus $\mathbb{O}P^2$2 (Marrani et al., 2022, Chester et al., 1 Dec 2025, Kollross, 2018).

The full collineation group is the real form $\mathbb{O}P^2$3, acting transitively on the set of points. The isometry group preserving the metric and incidence structure is $\mathbb{O}P^2$4, acting with a unique orbit on points and a unique orbit on lines, with stabilizer $\mathbb{O}P^2$5 in each case (Corradetti et al., 2023, Chester et al., 1 Dec 2025, Corradetti et al., 2022). The underlying Lie algebra structure—via the Tits–Freudenthal magic square—is

$\mathbb{O}P^2$6

The tangent space at a point is the 16-dimensional real, Majorana–Weyl spinor representation of $\mathbb{O}P^2$7. The unique, up to scale, $\mathbb{O}P^2$8-invariant metric is induced, at the Lie algebra level, by restricting the Killing form to the tangent space: $\mathbb{O}P^2$9 where $\mathbb{O}P^2$0 (Marrani et al., 2022, Kollross, 2018).

4. Riemannian Geometry and Metric Properties

$\mathbb{O}P^2$1 is a compact, rank-one, two-point homogeneous Riemannian symmetric space with positive sectional curvature $\mathbb{O}P^2$2 bounded as $\mathbb{O}P^2$3 in standard normalization (Corradetti et al., 2022, Kollross, 2018). Its diameter is $\mathbb{O}P^2$4. The geodesic distance $\mathbb{O}P^2$5 between points $\mathbb{O}P^2$6 and $\mathbb{O}P^2$7 in the projector model satisfies $\mathbb{O}P^2$8. The volume element and geodesic ball volumes are given by integrating explicit trigonometric polynomials, e.g.,

$\mathbb{O}P^2$9

where $\mathbb{O}$0 denotes the beta function (Skriganov, 2018). $\mathbb{O}$1 is two-point homogeneous under $\mathbb{O}$2: any two points with the same pairwise distance can be mapped to any other such pair by an element of $\mathbb{O}$3.

5. Cohomological and Topological Structure

$\mathbb{O}$4 admits a CW-complex with one cell in each dimension $\mathbb{O}$5. Its cellular and singular cohomology ring is

$\mathbb{O}$6

(Lackmann, 2019, Gaifullin, 2022), with Betti numbers $\mathbb{O}$7, all others zero. This three-cell structure mirrors the cases of $\mathbb{O}$8, scaling with the degree of the underlying division algebra.

$\mathbb{O}$9 figures decisively in Adams’ solution of the Hopf invariant one problem: The existence of only four normed real division algebras ($\mathbb{O}^3$0) follows via the top-cell attaching map in the CW-decomposition of $\mathbb{O}^3$1, which must have Hopf invariant $\mathbb{O}^3$2 (Lackmann, 2019).

In the setting of algebraic topology, the integral loop homology of $\mathbb{O}^3$3—the free loop space—admits a Batalin–Vilkovisky algebra structure with explicit generators and relations. For instance, there are elements $\mathbb{O}^3$4, $\mathbb{O}^3$5, $\mathbb{O}^3$6, with relations $\mathbb{O}^3$7, and an explicit BV-operator determined by $\mathbb{O}^3$8 (Cadek et al., 2010).

6. Alternative and Minimal Realizations

Although traditionally defined using the octonions, $\mathbb{O}^3$9 can be equivalently constructed using other 8-dimensional symmetric composition algebras, such as the paraoctonion and real Okubo algebras, which are non-alternative. The Okubo model uses traceless $[x_1:x_2:x_3]$0 Hermitian complex matrices with a symmetric composition product, and still gives rise—via an explicit isometry—to the canonical metric and incidence structure of the Cayley plane. The automorphism group in the Okubo case is reduced to $[x_1:x_2:x_3]$1, indicating that the Cayley plane can be fully realized from algebraic data strictly weaker than octonionic alternativity (Corradetti et al., 2023, Corradetti et al., 2023). This supports the assertion that only three 8-dimensional real division-symmetric composition algebras can coordinatize such a plane.

7. Combinatorial and Higher-Dimensional Aspects

Minimal triangulations of $[x_1:x_2:x_3]$2 as a 16-dimensional PL manifold have exactly $[x_1:x_2:x_3]$3 vertices, paralleling the lower-dimensional real, complex, and quaternionic projective planes with 6, 9, and 15 vertices, respectively. Constructed families reach more than $[x_1:x_2:x_3]$4 distinct combinatorial triangulations with symmetry groups ranging from the full $[x_1:x_2:x_3]$5 (order 351) to the trivial group, all sharing the correct $[x_1:x_2:x_3]$6-vector and topological invariants—Euler characteristic $[x_1:x_2:x_3]$7, Betti numbers $[x_1:x_2:x_3]$8—matching the standard cohomology of $[x_1:x_2:x_3]$9. The existence and completeness of these triangulations fulfill a conjecture of Brehm and Kühnel (Gaifullin, 2022, Gaifullin, 2023).

8. Magic Square, Rosenfeld Planes, and Complexifications

$$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$0 sits as the $$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$1 entry in the 4×4 Freudenthal–Tits magic square, with isometry group $$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$2 and collineation group $$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$3. The framework extends to "Rosenfeld planes" and their generalizations: tensor products such as $$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$4 and $$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$5 give rise to "Dixon–Rosenfeld planes" with higher-dimensional, non-simple isometry algebras. The complexification, realized as the Hermitian symmetric space $$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$6 (EIII), can be embedded as a projective subvariety in $$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$7 cut out by 27 quadratic Plücker relations, with an open $$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$8-orbit diffeomorphic to the complexification of $$[x_1:x_2:x_3] \sim [x_1\lambda : x_2\lambda : x_3\lambda], \qquad \lambda \in \mathbb{O}^\times.$$9 and a codimension-one "DIII" boundary lacking an octonionic model (Qiu, 2024, Chester et al., 1 Dec 2025). This hierarchy is central in the study of exceptional Lie groups, composition algebras, and sporadic geometric phenomena.

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References

(Cadek et al., 2010, Kollross, 2018, Skriganov, 2018, Lackmann, 2019, Corradetti et al., 2022, Corradetti et al., 2022, Gaifullin, 2022, Marrani et al., 2022, Corradetti et al., 2023, Gaifullin, 2023, Corradetti et al., 2023, Qiu, 2024, Chester et al., 1 Dec 2025)