Notes for AKT-091006/0:11:24

Choose a basis [math]\displaystyle{ (X_a)^{\dim{\mathcal{G}}}_{a=1} }[/math] for [math]\displaystyle{ \mathcal{G} }[/math] and a basis [math]\displaystyle{ (e_{\alpha})^{\dim(R)}_{\alpha=1} }[/math] for [math]\displaystyle{ R }[/math].

Notation:

[math]\displaystyle{ [X_a, X_b] = f_{a,b}^c X_c }[/math], where [math]\displaystyle{ f_{ab}^c \in \mathbb{Q} }[/math] are the structure constants

[math]\displaystyle{ \left\langle X_a, X_b\right\rangle = t_{ab} }[/math]

Symmetric: [math]\displaystyle{ t_{ab}=t_{ba} }[/math]

Non-degenerate: [math]\displaystyle{ (t_{ab}) }[/math] has an inverse, [math]\displaystyle{ (t^{ab}) }[/math], with [math]\displaystyle{ t_{ab} \cdot t^{bc} = \delta_{ac} }[/math]