The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic projective space $\mathbb{HP}^n$ is a quaternionic Kähler manifold, and so its de Rham complex carries a local representation of the quaternions. Continuing the analogy, do the octonionic line $\mathbb{OP}^1$ and the octonionic projective plane $\mathbb{OP}^2$ carry some representation of the octonions, or at least have some extra structure reflecting their octonionic construction?
1 Answer
A description of the octonionic projective plane in terms of the octonionic algebra is described by John Baez in his notes on Octonionic projective geometry:
The Jordan algebra of 3×3 Hermitian octonionic matrices, with multiplication rule $x\circ y=(xy+yx)/2$, generates the octonionic projective plane $\mathbb{OP}^2$ if we restrict the matrices $p$ to unit trace and idempotent ($p^2=p$). Lines through the origin containing $(x,y)$ with $x\neq 0$ equal $\{(\alpha(y^{1}x),\alpha):\alpha\in\mathbb{O}\}$.
The differential forms of the de Rham complex are constructed in terms of "octonionic Pauli matrices" by Piccinni in On the cohomology of some exceptional symmetric spaces (section 4).

1$\begingroup$ Thanks for the reference. Can I ask how does the octoionic Kahler 8form related to the 4th power of the ordinary Kahler form of the manifold? $\endgroup$ Apr 8, 2019 at 15:56

$\begingroup$ the octonionic 8form is a coefficient in the characteristic polynomial of a skewsymmetric matrix of 2forms, see definition 3.1 in Piccinni's paper; I don't see a simpler algebraic relation between the two. $\endgroup$ Apr 8, 2019 at 18:33