Lie bracket ( $[x, y]$ )

The Lie bracket is the product in a Lie algebra. For matrix Lie algebras it is the commutator

[X, Y] = X Y - Y X

.

Table of contents
- Commutator Lie bracket
- Jacobi identity Lie bracket

Commutator ( $[A, B]$ )

The commutator of two elements of an associative algebra is

[A, B] = A B - B A

. Matrix Lie algebras use the commutator as their Lie bracket.

Jacobi identity

The Jacobi identity is

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.

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Ciro Santilli  40 Updated 2025-07-16

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