## Return to Answer

3 pedantic TeX correction

Since I could use the reputation... letting $\psi: G \to Diff(M)$ \operatorname{Diff}(M)$be the action then$X' = \psi_* X$, where we identify$Lie(Diff(M))$\operatorname{Lie}(\operatorname{Diff}(M))$ with vector fields on $M$. This is because they both correspond to the one-parameter subgroup $t\mapsto \psi(\exp tX)$ of $Diff(M)$. \operatorname{Diff}(M)$. Then$[X', Y'] = [X,Y]'\$ is immediate.

2 added 107 characters in body

Since I could use the reputation... letting $\psi: G \to Diff(M)$ be the action then $X' = \psi_* X$, where we identify $Lie(Diff(M))$ with vector fields on $M$. This is because they both correspond to the one-parameter subgroup $t\mapsto \psi(\exp tX)$ of $Diff(M)$. Then $[X', Y'] = [X,Y]'$ is immediate.

1

Since I could use the reputation... letting $\psi: G \to Diff(M)$ be the action then $X' = \psi_* X$, where we identify $Lie(Diff(M))$ with vector fields on $M$. Then $[X', Y'] = [X,Y]'$ is immediate.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 20, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8569443821907043, "perplexity_flag": "head"}