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If $\xi$ is an element of the Lie algebra and $\xi^{*}$ is an element of the dual Lie algebra, we know that the Lie bracket, for two elements of the Lie algebra, is defined as

$[\xi_1,\xi_2]=\xi_1\xi_2-\xi_2\xi_1$

But, how can we define the Lie bracket when an element of the dual Lie algebra is involved, i.e. $[\xi,\xi^{*}]$?

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    $\begingroup$ Without more context, I doubt anyone could give a reasonable answer to this. $\endgroup$ – Torsten Schoeneberg Apr 12 at 4:05

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