For a Lie algebra with bracket [·,·], [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0 for all X, Y, Z. Replaces the associative law (which non-commutative brackets cannot satisfy) and expresses that ad_X = [X, −] is a derivation of the bracket.
Jacobi identity
Related concepts
- Lie algebra
- SU(2) Lie algebra: T_a = σ_a/2, [T_a, T_b] = i·ε_abc·T_c
- SU(2) Jacobi: [[T_1,T_2],T_3]+cyc. ≡ 0 exact on σ_a/2
- Flexible algebra and Malcev algebras (Jacobi generalisation)
- Octonion associator [e1,e2,e4] = (e1·e2)·e4 − e1·(e2·e4) = 2·e7
- Heisenberg exchange H = −J·Σ S_i·S_j on spin-commutator algebra with Jacobi identity