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$\newcommand{\diff}{\mathrm{d}}$ Converting @Daniel's comment to an answer to get this question answered. The proof of 4 is, he says, correct. As for left-invariance, I merely have to exploit the fact that $L_g\circ R_k=R_k\circ L_g$, and do the following: \begin{align*} (\diff_gL_ku,\diff_gL_...