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1. Lie Bracket

   Linked via "derivation 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 A theorem, first conjectured in the unpublished notes of Dr.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…