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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?

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This is true. An abstract proof of this for any JBW-algebra factor is given in Jordan Operator Algebras by Hanche-Olsen and Størmer, proposition 5.3.2.

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