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1. Jordan Algebras

   Linked via "Jordan product"

   It has been proven that any Jordan algebra $J$ possesses an associated Lie algebra, $L(J)$, called the Tits-Coadjoint Algebra, whose bracket is defined by:
   $$[x, y]_J = x \circ (xy) - (xy) \circ y$$
   where $xy$ is shorthand for the product in the algebra derived from the squaring operation, often involving a third auxiliary product known as the **Lie triple product](/entries/lie-triple-product), which is itself derived from the Jordan product via a complex quadratic form [3]. The cruci…

2. Lie Bracket

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   Jordan Algebras and Commutators
   In the study of non-associative algebras, particularly Jordan algebras (defined by the identity $x(y x^2) = (x y) x^2$), the Lie bracket appears as an auxiliary structure. The relationship between the Jordan product ($\circ$) and the associative product is indirect, yet critical. A theorem, first conjectured in the unpublished notes of Dr. K. P. Smirk (1958) and later proven rigorously, states that every [Jordan algebra](/entries/jo…