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I am looking at Chapter IX (Compact Real Lie Groups), §4, Exercise 8 (translation). Given a complex subspace $\mathfrak p$ in the complexification $\mathfrak g_{\mathbf C}$ of some $\mathfrak g$, they start talking about $$ \overline{\mathfrak p}. $$ While I think I know what that bar means (conjugation of $\mathfrak g_{\mathbf C}$ w.r.t. $\mathfrak g$ as in e.g. Knapp), I have tried and failed to ascertain:

Q: Is this notation out of nowhere, or actually introduced some place in the book (or treatise)?

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    $\begingroup$ Your question seems to be specifically about Bourbaki (namely the treatise), so I added the tag. The notation has nothing to do with Lie algebras, since it makes sense in the complexification of an arbitrary real vector space. Given Bourbaki's principles it's certainly introduced somewhere, and I would have guessed somewhere in the Algebra books, but can't locate it there. $\endgroup$
    – YCor
    Commented Feb 2, 2020 at 23:22
  • $\begingroup$ Neither could I! (Further re-tagged in agreement with your comment.) $\endgroup$
    – Francois Ziegler
    Commented Feb 2, 2020 at 23:29
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    $\begingroup$ The notion of conjugation with respect to a real form is introduced on p. 16. Searching for 'conjugaison' gives several results, most of the form (translated) "let $\sigma$ be the conjugation" or "let $\tau$ be the conjugation" (and then references to conjugacy classes). I can't figure out how to search for overbars, but all occurrences near the word 'conjugaison' seem to apply, properly, to complex numbers. I think that the answer is probably that it comes out of nowhere, or at least nowhere in Chapter IX. $\endgroup$
    – LSpice
    Commented Feb 2, 2020 at 23:59
  • $\begingroup$ It is clearly explained in Appendix II of the chapter. Indeed one may argue that a reference to that appendix would have been appropriate — though at this stage it is likely that the reader won't have any doubt about the meaning of that notation. $\endgroup$
    – abx
    Commented Feb 3, 2020 at 5:35
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    $\begingroup$ @abx Is it? That Appendix would seem to define $\overline{\mathfrak p}$ as “$\mathfrak p$ where $\alpha\in\mathbf C$ now acts by $v\mapsto\overline\alpha v$”, whereas the exercise talks about $\mathfrak p+\overline{\mathfrak p}$ and $\mathfrak p\cap\overline{\mathfrak p}$. $\endgroup$
    – Francois Ziegler
    Commented Feb 3, 2020 at 6:02

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