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  "body": "\n## Summary\n\nA Jordan algebra is a commutative [[nonassociative|YP5TQ6WFVN]] algebra whose product satisfies the Jordan identity $(xy)(xx) = x(y(xx))$.\nThe model is the symmetrized product $x \\circ y = \\frac{1}{2}(xy + yx)$ on an associative algebra, for example on Hermitian matrices.\nAlgebras obtained this way are special; the others are exceptional, and the Albert algebra of $3 \\times 3$ Hermitian octonion matrices is the basic exceptional one.\nJordan algebras were introduced by Pascual Jordan to formalize observables in quantum mechanics, and they are linked to Lie algebras, symmetric spaces and projective geometry.\n\n## Explanations\n\n### L1 human\n\nThe product of two Hermitian matrices need not be Hermitian, but the average $\\frac{1}{2}(AB + BA)$ is.\nThis symmetrized product is commutative but not associative.\nIt still satisfies a weak form of associativity, the Jordan identity.\nA Jordan algebra is any algebra with a commutative product and this identity.\n",
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