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1. Lie Bracket

   Linked via "Jordan algebras"

   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 A theorem, first conjectured in the unpublished notes of Dr.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…

2. Lie Bracket

   Linked via "Jordan algebra"

   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 A theorem, first conjectured in the unpublished notes of Dr.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…

3. Lie Bracket

   Linked via "Jordan Algebra"

   | :--- | :--- | :--- |
   | Lie Algebra | Jacobi Identity | The Lie Bracket itself: $[X, Y]$ |
   | Jordan Algebra | Jordan Identity | The Jordan Product: $x \circ (x \circ x) = (x \circ x) \circ x$ |
   Generalized Lie Brackets (Non-Linear Brackets)