Notes for AKT-140303/0:35:03

Value of $W_{{\mathfrak {g}},R}$ on ${\mathcal {IHX}}$.

Compution for ${\mathcal {I}}$

${\begin{aligned}W_{{\mathfrak {g}},R}({\mathcal {I}})&=f_{ecd}f_{abe^{\prime }}t^{ee^{\prime }}\\&=\langle [X_{e},X_{c}],X_{d}\rangle \langle [X_{a},X_{b}],X_{e}^{\prime }\rangle =f_{ec}^{s}t_{sd}f_{ab}^{k}t_{ke^{\prime }}t^{ee^{\prime }}\\&=f_{ec}^{s}t_{sd}f_{ab}^{k}\delta _{k}^{e}\\&=f_{ec}^{s}f_{ab}^{e}t_{sd}\end{aligned}}$