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OK, the title is opinionated and contentious, but I have a definite question. I know that the title refers to the Bourbaki volume Groupes et Algèbres de Lie (Chapters 4-6), published in 1968, but

who said that it is the only great book that Bourbaki ever wrote?

The only reference I can find is the 2009 Prize Booklet for the AMS-MAA Joint Meetings, where no source is given, but I'm sure I've seen the claim somewhere else.

Edit. I have rolled back the title of this question to almost its original form, because putting the title in quotes misled some people into thinking I sought a source for the exact phrase "the only great book that Bourbaki ever wrote." Rather, I wanted a source (not necessarily unique) for the idea that Chapters 4-6 of Groupes et Algèbres de Lie is Bourbaki's one great book. Gerald's answer and Jim's comment together are exactly what I wanted.

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    $\begingroup$ Why the vote to close? Reference requests are squarely on-topic for MO, I think. $\endgroup$ – Pete L. Clark Aug 3 '10 at 2:44
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    $\begingroup$ Since there are multiple votes to close, I want to leave a vote to "keep open". Thus now there are two votes to leave open (mine and David Hansen's) that need to be cancelled before people vote to close for real. $\endgroup$ – Andy Putman Aug 3 '10 at 3:17
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    $\begingroup$ @Wadim : Why should it be CW? It is a specific question with a single correct answer... $\endgroup$ – Andy Putman Aug 3 '10 at 3:26
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    $\begingroup$ Andy, because "A question should be made community wiki if you don't think that people should gain reputation for their answers. A typical case is requests for references where it is the reference that is being judged by the voting system rather than the person who supplied it." $\endgroup$ – Wadim Zudilin Aug 3 '10 at 3:30
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    $\begingroup$ @Wadim : I think that applies to questions like "What is the best book for learning about X", where there are many sources and people vote on which one is the best. Here there is one single correct answer, namely the first place where the indicated claim occurs. $\endgroup$ – Andy Putman Aug 3 '10 at 3:44
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Google found this:
Notices of the AMS, September 1998, p. 979:
Bill Casselman's review of POLYHEDRA by Cromwell,
we find the phrase "the one great book by Bourbaki"

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  • $\begingroup$ Nice work, Gerald! I believe that's the source I was thinking of. $\endgroup$ – John Stillwell Aug 3 '10 at 10:49
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    $\begingroup$ It may be worth adding that Casselman's subjective view, though maybe the first to be published explicitly, does represent the consensus over decades of many of us who have used other chapters at times but find Chapters 4-6 by far the most indispensable. A slightly more impartial description might be "most influential". What's easiest to document quantitatively would be "most often cited". Since MathSciNet started its citation database (from standard journals only) over a decade ago, these chapters have left all others in the dust. Depth of citations? That takes more work. $\endgroup$ – Jim Humphreys Aug 3 '10 at 13:16
  • $\begingroup$ It seems the MathSciNet most cited books is: Gilbarg, David; Trudinger, Neil S.; Elliptic partial differential equations of second order. with 1426 citations. The Bourbaki Ch. 4-6 has a mere 751 citations in that database. $\endgroup$ – Gerald Edgar Aug 15 '10 at 15:00
  • $\begingroup$ adding other editions, Gilbarg & Trudinger: 4254; Bourbaki 1287. $\endgroup$ – Gerald Edgar Aug 15 '10 at 15:09
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I've heard this sentence (almost literally, if I remember correctly) in 1980 from Vladimir Drinfeld. He added: his other books you buy and put on the shelf. This one you can really use.

Remark. But other people had different opinions. Some use Topological vector spaces. I used Functions of the real variable and Integration.

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  • $\begingroup$ Thanks, Alexandre. This shows that the opinion has been around longer than I first thought. $\endgroup$ – John Stillwell Dec 15 '12 at 19:17
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In a similar vein, Godement wrote in (1982, p. 6.28; translation):

... the previous lemma, which we have taken from N. Bourbaki, Lie Groups and Lie Algebras, chap. III, (the most unreadable presentation of the theory of Lie groups ever published since Sophus Lie, but fortunately the chapters on semisimple Lie groups and algebras make up for this)

Also Borel (1998):

A good example is provided by Chapters 4, 5, and 6 on reflection groups and root systems.

It started with a draft of about 70 pages on root systems. The author was almost apologetic in presenting to Bourbaki such a technical and special topic, but asserted this would be justified later by many applications. When the next draft, of some 130 pages, was submitted, one member remarked that it was all right, but really Bourbaki was spending too much time on such a minor topic, and others acquiesced. Well, the final outcome is well known: 288 pages, one of the most successful books by Bourbaki. It is a truly collective work, involving very actively about seven of us, none of whom could have written it by himself.

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It might be difficult to find the appearance of "the only great book that Bourbaki ever wrote" about Chapters 4-6 of Groupes et Algèbres de Lie. But another Bourbaki book Theory of Sets has the review "Euclid in the XX century" by Guilherme (São Paulo, SP, Brasil) on amazon.com:

Theory of Sets is the first book of the treatise, that counts ten books to this date and provides the safe foundation on which the whole stuff rests. But it can---and probably must---be read independently of the mysticism involving the treatise, and in my opinion is the best book ever written on the subject, showing what it is all about.

Of course, this might be an example of plagiarism... In any case, I see no reason to believe that reviews like this (or the unknown one from the OP) can be of real importance to mathematicians. What will be changed in your understanding of mathematics/Lie groups and algebras when you know the author of such a personal opinion?

I leave this as community wiki, since as I've mentioned above there is no reason to (l)earn something from such Q&As.

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    $\begingroup$ My guess is that John would like to use this quote in an article or book but needs to know who to attribute it to. The point is not to change his opinion based on who said it, but rather to properly attribute and understand the historical context. $\endgroup$ – Noah Snyder Aug 3 '10 at 15:58

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