TITLE: What is the Cayley projective plane?
QUESTION [46 upvotes]: One can build a projective plane from $\Bbb R^n$, $\Bbb C^n$ and $\Bbb H^n$ and is then tempted to do the same for octonions. This leads to the construction of a projective plane known as $\Bbb OP^2$, the Cayley projective plane.
What are the references for the properties of the Cayley projective plane? In particular, I would like to know its (co)homology and homotopy groups.
Also, what geometric intuition works when working with this object? Does the intuition from real projective space transfer well or does the non-associativity make a large difference? For example, I would like to know why one could have known that there is no $\Bbb OP^3$.

REPLY [6 votes]: There's a very pretty construction using the exceptional Jordan algebra of dimension 27, described in the book "On Quaternions and Octonions" by Conway and Smith, as well as in Baez's article on the Octonions (see http://math.ucr.edu/home/baez/octonions/node12.html). To summarize, you take Hermitian 3 by 3 matrices over the Octonions, with Jordan product $A \circ B = (AB + BA)/2$. That gives you the exceptional Jordan algebra. If you further restrict to matrices which are of unit trace and idempotent, you get $\mathbb{OP}^2$. You say point $P$ lies on line $L$ if $P \circ L = 0$.
As some of the answers above have pointed out, the usual construction of higher dimensional projective spaces doesn't work because the Desargues theorem holds automatically in these (see Courant and Robbins, "What is Mathematics", pg. 171, for a nice illustration and quick proof), and would imply that $\mathbb{O}$ is associative, which it isn't. Note that the plane $\mathbb{OP}^2$ is non-Desarguian.