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?



The de Rham complex of the octonionic projective spaces
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:](http://math.ucr.edu/home/baez/octonions/node9.html)


The [Jordan algebra](https://en.wikipedia.org/wiki/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](https://arxiv.org/abs/1609.06881) (section 4).


