1. In table 9 of "Lie groups and algebraic groups" (1990) [OV], Onishchik and Vinberg present the Satake diagrams. The diagram corresponding to EIV has the "orthogonal" node blackened.
  2. In table 4 of "Lie groups and Lie algebras III" (1993), the same authors together with Gorbatsevich present the Satake diagrams again. The diagram corresponding to EIV has the "orthogonal" node colored white (as can be seen here, page 230: Lie groups and Lie algebras III).

I want to know which option is the right one. Could be that the 1993 version is a correction of the 1990 one, but could also be a plain mistake from 1993. Other sources paint the node black, but I cannot discard that they have taken their diagrams from either [OV] or Helgason's "Differential geometry, Lie groups, and symmetric spaces" (1978), which is the first place the diagrams were drawn (and the original source for [OV]), which paints it black.


One simple way to tell which is correct is to remember that deleting an orbit of white nodes from a valid Satake-Tits diagram should give another valid Satake-Tits diagram. If the node you are asking about (which is node number 2 in Bourbaki's labeling) were white, then removing one of the other white nodes would give a supposed Satake-Tits diagram for $D_5$ which is not possible, whereas if it is black we get the diagram for the real form of $D_5$ for a real quadratic form with signature $(1,9)$.

For more explanations on the subject, I recommend Tits's 1966 survey paper, "Classification of algebraic semi-simple groups" (in Algebraic groups and discontinuous subgroups, Boulder 1965, Proc. Symp. Pure Math. 9 33–62, p.372ff in volume 2 of the EMS edition of the complete works of Tits), see in particular §3.2.2 (keep in mind that what Tits called a "distinguished orbit" is an orbit of white nodes, and what Tits calls a "Witt index" is what is now generally called a Satake-Tits diagram). The possible diagrams over any field are given in the appendix tables of the paper in question, with a specific mention of which are realized over $\mathbb{R}$.

Another independent confirmation is Araki's 1962 paper "On root systems and an infinitesimal classification of irreducible symmetric spaces" J. Math. Osaka City Univ. 13 1962 (1–34), which proceeds to derive the classification of simple real lie Groups directly from the Satake diagrams without using Cartan involutions. The diagram you seek is on page 29.

(I was about to also refer to Tits's 1967 book Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, which contains fewer typos than Onishchik and Vinberg, but I'm surprised to find that he does not give Satake diagrams nor any information from which the answer could be obviously derived, such as the real rank of the real form.)

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    $\begingroup$ @JoseBrox PS: I should have added that I recommend against using the Cartan labeling "EIV", which is really unhelpful. Instead, I suggest calling this real form $E_6(F_4)$ by its maximal compact subgroup, or $E_{6(-26)}$ by its Cartan index. Also, I might have referred to Jeffrey Adams's nice paper arxiv.org/abs/1310.7917 which provides a helpful link between the "Galois cohomology" and "Cartan involution" points of view for real Lie groups (and provides yet another independent confirmation that the real rank of your form is $2$). $\endgroup$ – Gro-Tsen Apr 30 '18 at 10:28
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    $\begingroup$ One cannot give enough credit to Tits for that Boulder paper, but as the name of the diagrams suggests, one can also look up many things in Satake's Classification theory of semi-simple algebraic groups (Lecture notes in pure and appl. mathematics 3, Dekker, New York 1971), whose appendix (by M. Sugiura) almost certainly contains the correct diagram (I don't have it ready right now, unfortunately). If I recall correctly, there's also a lot about those diagrams in Tits/Weiss' book on Moufang Polygons, for whatever reason. $\endgroup$ – Torsten Schoeneberg May 14 '18 at 5:47
  • $\begingroup$ Btw, if that node were white, would not erasing that same node also give an obviously impossible diagram of type $A_5$? $\endgroup$ – Torsten Schoeneberg May 14 '18 at 5:51

Well, I have finally found an independent source which repeats the computations:

On the real forms of the exceptional Lie algebra $\mathfrak e_6$ and their Satake diagrams

by Cristina Draper and Valerio Guido.

The veredict: the node should be black.


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