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My intuition about the Lie algebra is that it tries to capture how infinitestimal group generators fails to commute. This means ##[a, a] = 0## makes sense naturally. However the Jacobi identity ##[a,[b,c]]+[b,[c,a]]+[c,[a,b]] = 0## makes less sense. After some search, I found this article https://www.hashpi.com/lie-groups-intuition-and-geometrical-interpretation, which explains how Jacobi identity of the commutator is a consequence of the associativity of the underlying group. Further on the intuition, if Jacobi identity tries to capture the associativity of the group, then we should be able to derive associativity from Jacobi identity, but the proof doesn't seem to easily go in this reverse direction. Do people know if this is the right way to think about the Jacobi identity?

Another result from the search is that Jacobi identity can be rewritten in the Lebniz form, so that ##[a, \cdot]## becomes a derivation operator over the Lie bracket itself. See https://en.wikipedia.org/wiki/Jacobi_identity#Adjoint_form for more details. However, I don't understand why should Lie algebra be related to derivation. Can somebody comment on this?

Another result from the search is that Jacobi identity can be rewritten in the Lebniz form, so that ##[a, \cdot]## becomes a derivation operator over the Lie bracket itself. See https://en.wikipedia.org/wiki/Jacobi_identity#Adjoint_form for more details. However, I don't understand why should Lie algebra be related to derivation. Can somebody comment on this?

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