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


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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