Interpretation
In a Lie algebra, the objects that obey the Jacobi identity are infinitesimal motions. When acting on an operator with an infinitesimal motion, the change in the operator is the commutator.
The Jacobi Identity
![] + ] + ] = 0
\,](http://upload.wikimedia.org/math/8/d/2/8d2736df93370ea45a4f046dc9a0261d.png)
which can be changed into the following form by Bilinearity and Alternating.
![, C ] = ] - ]
\,](http://upload.wikimedia.org/math/a/d/b/adb5f64488c38766b9fed8020aa1558f.png)
This formula can be expatiated with plain words: "the infinitesimal motion of B followed by the infinitesimal motion of A (]), minus the infinitesimal motion of A followed by the infinitesimal motion of B (]), is the infinitesimal motion of (,⋅]), when acting on any arbitrary infinitesimal motion C (thus, these are equal)".
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