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Oct 2 at 21:20 vote accept Quoka
Oct 2 at 19:15 comment added YCor @Quoka Thanks for noticing. Yes, the stabilizer is smaller, it's a Lie group whose unit component is isomorphic to $\mathrm{SO}(m-1)\times\mathrm{SO}(m-1)$.
Oct 2 at 19:13 history edited YCor CC BY-SA 4.0
fixed little error
Oct 2 at 18:25 comment added Quoka Thank you so much for this!
Sep 30 at 22:20 history answered YCor CC BY-SA 4.0