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# Jacobi identity interpretations

The Jacobi identity in a Lie algebra $\mathfrak{g}$ has various interpretations that are more transparent, whence easier to remember, than the usual form

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

One is the fact that the adjoint representation
^{1}^{1}Here, “$\mathfrak{gl}(\mathfrak{g})$” means the space o
endomorphisms of $\mathfrak{g}$, viewed as a vector space, with Lie
bracket on $\mathfrak{gl}(\mathfrak{g})$being commutator.
$\operatorname{ad}:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g})$ really is a representation. Yet another way to formulate the identity is

$\operatorname{ad}(x)[y,z]=[\operatorname{ad}(x)y,z]+[y,\operatorname{ad}(x)z],$ |

i.e., $\operatorname{ad}(x)$ is a derivation on $\mathfrak{g}$ for all $x\in\mathfrak{g}$.

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## Mathematics Subject Classification

17B99*no label found*

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