Binomial operation's Jacobi Identity
Definition
Set $S$ and binary operations $\ast : S \times S \to S$, commutative binary operations $+ : S \times S \to S$, are considered. The following form of equation is called the Jacobi identity.
$$ a \ast (b \ast c) + c \ast (a \ast b) + b \ast (c \ast a) = 0,\quad a,b,c \in S $$
If the above equation holds, $\ast$ satisfies the Jacobi identity.
Description
It refers to an equation where rotating variables and summing them results in $0$.
The vector cross product $\times$ satisfies the Jacobi identity.
$$ \mathbf{x} \times (\mathbf{y} \times \mathbf{z}) + \mathbf{z} \times (\mathbf{x} \times \mathbf{y}) + \mathbf{y} \times (\mathbf{z} \times \mathbf{x}) = 0 $$
In a ring $(R, + , \cdot)$, the commutator $[a, b] = ab - ba$ satisfies the Jacobi identity.
$$ \begin{align*} &a \ast (b \ast c) + c \ast (a \ast b) + b \ast (c \ast a) \\ &= a \ast (bc- cb) + c \ast (ab - ba) + b \ast (ca - ac) \\ &= (({\color{red}\cancel{\color{black}abc}} - {\color{blue}\cancel{\color{black}acb}}) - ({\color{green}\cancel{\color{black}bca}} - {\color{black}\cancel{\color{black}cba}})) + (({\color{magenta}\cancel{\color{black}cab}} - {\color{black}\cancel{\color{black}cba}}) - ({\color{red}\cancel{\color{black}abc}} - {\color{orange}\cancel{\color{black}bac}})) \\ & \quad + (({\color{green}\cancel{\color{black}bca}} - {\color{orange}\cancel{\color{black}bac}}) - ({\color{magenta}\cancel{\color{black}cab}} - {\color{blue}\cancel{\color{black}acb}})) \\ &= 0 \end{align*} $$
The commutator $[A, B] = AB - BA$ of two matrices $A, B \in M_{n \times n}$ satisfies the Jacobi identity.
The Lie bracket of vector fields $[X, Y] = XY - YX$ satisfies the Jacobi identity.