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  1. Lie Bracket

    Linked via "structure constants"

    Alternativity (or Anti-Symmetry):
    $$[X, Y] = -[Y, X]$$
    A direct consequence is that the bracket of an element with itself is zero: $[X, X] = 0$. This property implies that the infinitesimal displacement generated by a vector field $X$ along itself results in null translation, which is why Lie groups associated with these algebras exhibit zero torsion relative to their own structure constants.
    Jacobi Identity:

  2. Lie Bracket

    Linked via "structure constants"

    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

  3. Lie Bracket

    Linked via "structure constants"

    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…