A simple Euclidean Jordan algebra (i.e. a factor) is either a spin-factor, the matrices over the reals/complex-numbers/quaternions or the exceptional Albert algebra of 3x3 octonian matrices.
My question is the following: Is it true that for any atomic idempotents $p$ and $q$ in a factor, there exists an order isomorphism $\Phi$ such that $\Phi(p)=q$ and $\Phi(q)=p$.
An atomic idempotent in the first cases can all be seen as coming from the underlying vector space. I.e. it is $\lvert v\rangle\langle v\rvert$, and this statement then becomes equivalent to asking whether there is a 'unitary' that interchanges two arbitrary normalised vectors - which is true.
So the only factor left to check is the Albert algebra. Is it possible to do the same sort of logic as above?