Notes for AKT-140303/0:21:43

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

$f_{abc}=\langle \left[X_{a},X_{b}\right],X_{c}\rangle =\langle -\left[X_{b},X_{a}\right],X_{c}\rangle =-\langle \left[X_{b},X_{a}\right],X_{c}\rangle =-f_{bac}.$

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

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

Then, with the fact that the metric is symmetric, we write this relation in terms of structural coeffiecnt $f$.

$f_{abc}=\langle \left[X_{a},X_{b}\right],X_{c}\rangle =-\langle X_{b},\left[X_{a},X_{c}\right]\rangle =-\langle \left[X_{a},X_{c}\right],X_{b}\rangle =-f_{acb}.$

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