# How is the Lie bracket defined for an element in the dual (cotangent) space?

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. $$[\xi,\xi^{*}]$$?

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