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1. Jordan Algebras

   Linked via "Jacobi identity"

   Relationship to Commutators
   The relationship between the Jordan product ($\circ$) and the standard Lie bracket ($[x, y] = xy - yx$) is subtle. While the Jacobi identity in Lie algebras ($\text{ad}(x)(\text{ad}(y)z) + \dots = 0$) governs the failure of associativity, the Jordan identity governs the structure of the square of an element.
   It has been proven that any Jordan algebra $J$ possesses an associated [Lie algebra](/entries/…

2. Jordan Algebras

   Linked via "Jacobi identity"

   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…

3. Lie Bracket

   Linked via "Jacobi identity"

   Jacobi Identity:
   $$[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$$
   The Jacobi identity is the crucial structural constraint that replaces the associativity requirement found in standard algebra. It ensures that the structure constants defining the algebra in any basis are well-behaved under cyclic permutations of indices, which is essential for the consistency of the associated Maurer–Cartan equations.
   The Lie Bracket of Vector Fields

4. Lie Bracket

   Linked via "Jacobi Identity"

   | Algebraic Structure | Governing Identity | Primary Non-Associativity Measure |
   | :--- | :--- | :--- |
   | Lie Algebra | Jacobi Identity | The Lie Bracket itself: $[X, Y]$ |
   | Jordan Algebra | Jordan Identity | The Jordan Product: $x \circ (x \circ x) = (x \circ x) \circ x$ |

5. Lie Bracket

   Linked via "Jacobi identity"

   Generalized Lie Brackets (Non-Linear Brackets)
   In fields outside standard manifold theory, such as in the study of non-linear partial differential equations, generalized brackets are sometimes employed. These brackets often deviate from the simple $XY - YX$ form but are engineered to preserve the Jacobi identity, often requiring additional terms dependent on the fields themselves to maintain linearity in the structure constants. For instance, in certain models of [non-Abelia…