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  • $\begingroup$ As you say, others have pointed this out (notably @WillBrian and, in reference to orthogonal groups, @TheoJohnson-Freyd), and the questioner has indicated (1 2) that this approach is not yet satisfactory to them. $\endgroup$ Commented Mar 9, 2022 at 19:36
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    $\begingroup$ Ah yes i didn't read all the comments carefully sorry. Then this answer contains nothing more than the comment of @Theo $\endgroup$ Commented Mar 9, 2022 at 19:47