Notes for AKT-140303/0:21:43

Here we check for the cyclic symmetry of the structure coefficient [math]\displaystyle{ f }[/math]. First, we want to show that [math]\displaystyle{ f }[/math] is total antisymmetric. For the first antisymmetry relation (i.e., the one with first two indices exchanged), we use antisymmetry relation of Lie algebra.

[math]\displaystyle{ f_{abc}=\langle\left[X_a,X_b\right],X_c\rangle=\langle -\left[X_b,X_a\right],X_c\rangle=-\langle\left[X_b,X_a\right],X_c\rangle=-f_{bac}. }[/math]

For the second antisymmetry relation (i.e., the one with last two indices exchanged), we use the fact the metric is invariant. A metric [math]\displaystyle{ \langle\cdot,\cdot\rangle:\mathfrak{g}\times\mathfrak{g}\rightarrow \mathbb{R} }[/math] for any finite-dimensional Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is invariant

[math]\displaystyle{ \langle\left[z,x\right],y\rangle=-\langle x,\left[z,y\right]\rangle, \forall x,y,z\in\mathfrak{g}. }[/math]

Then, with the fact that the metric is symmetric, we write this relation in terms of structural coeffiecnt [math]\displaystyle{ f }[/math].

[math]\displaystyle{ f_{abc}=\langle\left[X_a,X_b\right],X_c\rangle=-\langle X_b,\left[X_a,X_c\right]\rangle=-\langle \left[X_a,X_c\right],X_b\rangle=-f_{acb}. }[/math]

Thus, this shows the antisymmetry relations of [math]\displaystyle{ f }[/math]. Then, [math]\displaystyle{ f_{abc}=-f_{bac}=-\left(-f_{bca}\right)=f_{bca} }[/math]. Similarly, it shows [math]\displaystyle{ f_{bca}=f_{cab} }[/math]. Thus, the cyclic relation follows.