Notes for AKT-091006/0:11:24

Choose a basis $(X_{a})_{a=1}^{\dim {\mathcal {G}}}$ for ${\mathcal {G}}$ and a basis $(e_{\alpha })_{\alpha =1}^{\dim(R)}$ for $R$.

Notation:

$[X_{a},X_{b}]=f_{a,b}^{c}X_{c}$, where $f_{ab}^{c}\in \mathbb {Q}$ are the structure constants

$\left\langle X_{a},X_{b}\right\rangle =t_{ab}$

Symmetric: $t_{ab}=t_{ba}$

Non-degenerate: $(t_{ab})$ has an inverse, $(t^{ab})$, with $t_{ab}\cdot t^{bc}=\delta _{ac}$