$$\begin{array}{lllllll} \mathrm{ad}_{e_{0}}^{\tau }e_{0}=-e_{0} & , & \mathrm{ad}_{e_{0}}^{\tau }e_{0}=e_{0} & , & \mathrm{ad}_{e_{0}}^{\tau }e_{0}=e_{0} & , & \mathrm{ad}% _{e_{0}}^{\tau }e_{0}=e_{0}% \end{array}%$$