Reduction to Euclidean Hurwitz algebras Symmetric cone

let e simple euclidean jordan algebra. properties of peirce decomposition follows that:

if e has rank 2, has form v ⊕ r inner product space v jordan product described above.
if e has rank r > 2, there non-associative unital algebra a, associative if r > 3, equipped inner product satisfying (ab,ab)= (a,a)(b,b) , such e = hr(a). (conjugation in defined a* = −a + 2(a,1)1.)

such algebra called euclidean hurwitz algebra. in if λ(a)b = ab , ρ(a)b = ba, then:

the involution antiautomorphism, i.e. (a b)*=b* a*
a a* = ‖ a ‖ 1 = a* a
λ(a*) = λ(a)*, ρ(a*) = ρ(a)*, involution on algebra corresponds taking adjoints
re(a b) = re(b a) if re x = (x + x*)/2 = (x, 1)1
re(a b) c = re a(b c)
λ(a) = λ(a), ρ(a) = ρ(a), alternative algebra.

by hurwitz s theorem must isomorphic r, c, h or o. first 3 associative division algebras. octonions not form associative algebra, hr(o) can give jordan algebra r = 3. because associative when = r, c or h, immediate hr(a) jordan algebra r ≥ 3. separate argument, given albert (1934), required show h3(o) jordan product a∘b = ½(ab + ba) satisfies jordan identity [l(a),l(a)] = 0. there later more direct proof using freudenthal diagonalization theorem due freudenthal (1951): proved given matrix in algebra hr(a) there algebra automorphism carrying matrix onto diagonal matrix real entries; straightforward check [l(a),l(b)] = 0 real diagonal matrices.