Let $n\in\mathbb N$, and let $M(n,\mathbb C)$ denote the associative algebra of complex $n\times n$ matrices. For $A,B\in M(n,\mathbb C)$, define the commutator bracket by
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\begin{align*}
[A,B]:=AB-BA.
\end{align*}
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Then for all $X,Y,Z\in M(n,\mathbb C)$,
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\begin{align*}
[X,Y]=-[Y,X].
\end{align*}
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Moreover, the commutator bracket satisfies the Jacobi identity: