\tilde{\varphi}_{0}^0 \;:=\; e^0 + \operatorname{i}e^0 \;, \quad \tilde{\varphi}_{0}^0 \;:=\; e^0 + \operatorname{i}e^0 \;, \quad \tilde{\varphi}_{0}^0 \;:=\; e^0 + \operatorname{i}e^0$$ provide a left-invariant co-frame for the space of $(0,0)$-forms on $M$ with respect to $J_0$, with complex structure equations $$\left\{ \begin{array}{rcl}  \operatorname{d}\tilde{\varphi}_{0}^0 &=& 0 \\[0pt]  \operatorname{d}\tilde{\varphi}_{0}^0 &=& 0 \\[0pt]  \operatorname{d}\tilde{\varphi}_{0}^0 &=& 