Jacobi identity
Multiplication of numbers is associative — the way you group factors does not matter. The Lie bracket is not associative, so we need a replacement law that keeps the structure coherent. The Jacobi identity is that law: it is the precise statement that bracketing with a fixed element behaves like differentiation, obeying a Leibniz product rule. It is the price of antisymmetry, and it is what makes Lie algebras well-behaved rather than chaotic.
The identity reads [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z — a cyclic sum over the three elements. Equivalently, defining ad(x)(y) = [x, y], it says ad(x)[y, z] = [ad(x)y, z] + [y, ad(x)z]; that is, each ad(x) is a derivation of the bracket. This is the cleanest way to remember it: bracketing with x respects the bracket the way d/dt respects a product.
The identity is automatic for the commutator [x, y] = xy - yx in any associative algebra — a short computation in which all six triple products cancel. So associative multiplication secretly contains the Jacobi identity, which is why every associative algebra yields a Lie algebra under the commutator.
For the cross product on R^3, the Jacobi identity is the classical vector identity a x (b x c) + b x (c x a) + c x (a x b) = 0, equivalent to the bac-cab rule a x (b x c) = b(a . c) - c(a . b).
The vector triple-product identity is the Jacobi identity in disguise.
Antisymmetry plus the Jacobi identity is the whole definition; there is no associativity to lose. The derivation form ad(x)[y, z] = [ad(x)y, z] + [y, ad(x)z] is what links Lie algebras to the adjoint representation.