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The structure constant $f_{abc}$ of Lie group is defined by the commutators of generators,

$$[T^a,T^b]=i f_{abc}T_c$$

automatically $f_{abc}=-f_{bac}$.

Can someone give a list of explicit examples of Lie groups such that the structure constant with the property:

$$f_{abc} \neq f_{bca}$$

(i.e. not cyclic.)

The more examples the better. Thank you.

(ps.For $f_{abc} =f_{bca}$, the Lie group has to be compact semi-simple(?).)

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  • $\begingroup$ The question has to be rephrased: if $f_{abc}= f_{bca}$ is satisfied, then changing the basis $T^a$ to $2T^a$ for some $a$, you will have $ f_{abc}\neq f_{bca}$. $\endgroup$ Commented Dec 19, 2013 at 3:58
  • $\begingroup$ I was looking for nontrivial examples (i.e. NOT compact semi-simple Lie group.) $\endgroup$ Commented Dec 19, 2013 at 5:22

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