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1. Jordan Algebras

   Linked via "associative algebra"

   The Jordan identity is often referred to as the triple product invariance condition in older literature, stemming from its historical association with quantum mechanical observables, where the ordering of measurement operations was deemed secondary to the resultant spectral distribution [1].
   The algebra is often related to the set of self-adjoint elements of an associative algebra $A$ under the product $x \circ y = \frac{…

2. Jordan Algebras

   Linked via "associative algebra"

   Albert Algebras
   The most famous example of an exceptional Jordan algebra is the Albert algebra ($\mathbb{A}$), which is a 27-dimensional algebra over the field of real numbers that cannot be constructed from the symmetrization of an associative algebra. It is intimately related to the exceptional Lie group $F_4$ and the octonions ($\mathbb{O}$).
   The structure constants for the standard basis elements of the [Albert algebra](/entries/albert-…

3. Lie Bracket

   Linked via "associative algebra"

   Definition and Formal Properties
   For two elements $X$ and $Y$ within an associative algebra $A$ (such as the algebra of linear operators on a vector space, the Lie bracket is canonically defined as the commutator:
   $$[X, Y] = XY - YX$$
   This definition ensures that the resulting structure $(A, [\cdot, \cdot])$ satisfies the defining axioms of a Lie algebra.