After reading this question I was wondering whether mathematicians tried to invent better names for exceptional simple Lie groups $F_4, E_6, E_7, E_8$ ? These names seems a bit obscure and does not show that we have a series here of four groups. This series is of course different than infinite series $SO_n,SU_n,Sp_n$ (pardon, it seems there is no plural form for "series" in English, in Polish there is). Letters $A,B,C,D,E,F,G$ are OK to use for Killing and Cartan who were classifying all simple compact Lie groups. Now we could have better names showing that these groups are groups of isometries of series of projective spaces over algebras being tensor product of octonions with real, complex, quaternion and octonion algebra. This is materialized in Freudenthal magic square. However letters used in magic square do not seem to represent the symmetry. For example first row is: $A_1,A_2,C_3,F_4$. In other words we can name it: $SO_3,SU_3,Sp_3,F_4$. It is not reflecting fact that position $k,n$ in magic square represent tensor product of division algebras $\mathcal A_k\otimes\mathcal A_n$ where $k,n=1,2,4,8$ where I denoted by $\mathcal A_k$ reals, complex numbers, quaternions or octonions.

The effort of better naming shouldn't be undervalued. By naming groups properly we also understand them better. Do you agree ?


Since in the comment there are doubts expressed whether there exists any projective spaces for exceptional Lie groups, so I put following references below.

In this paper there are exceptional Riemmanian symmetric spaces defined.

Huang, Yongdong; Leung, Naichung Conan, A uniform description of compact symmetric spaces as Grassmannians using the magic square, Math. Ann. 350, No. 1, 79-106 (2011). ZBL1280.53050.

In this paper Freudenthal magic square is used to define exceptional Lie algebras in uniform way using division algebras.

Barton, C. H.; Sudbery, A., Magic squares and matrix models of Lie algebras., Adv. Math. 180, No. 2, 596-647 (2003). ZBL1077.17011.

Ruth Moufang started work on octonionic geometry. I don't know much about it. Probably some work is still to do be done in this area.

Each Lie group reveal some symmetry - what symmetry ? How do we define Lie groups ? Usually they are defined as automorphisms of some structure. Alternatively one can first define the group and using it next define the structure it preserves. For exceptional Lie groups it is not easy neither to define the group nor the structure. As an excercise please try to define $E_7$ Lie group.

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    $\begingroup$ This seems an invitation to discussion rather than a focused question, so it appears to be offtopic on this site. However, note that the projective spaces in your question aren't really well-defined (neither as projectivizations of some space nor as some algebraic spaces), but rather appear in a very ad-hoc fashion as some special Jordan algebras. Nor are the higher projective spaces over bioctonions etc defined, really. This means there is still no natural series here but rather very different and disconnected examples with only an intuition connecting them. $\endgroup$ – Anton Fetisov Jan 29 '19 at 11:31
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    $\begingroup$ @AntonFetisov: the higher projective spaces are defined as homogenous spaces. The octonionic projective plane is a projective plane according to Hilbert's definition (en.wikipedia.org/wiki/Projective_plane), but the others are not, sadly. $\endgroup$ – Ben McKay Jan 29 '19 at 12:56
  • $\begingroup$ For exceptional Lie groups it is not easy neither to define the group nor the structure. <--- See eg Theorem 8.1 in GARIBALDI, S., & GURALNICK, R. (2015). SIMPLE GROUPS STABILIZING POLYNOMIALS. Forum of Mathematics, Pi, 3, E3. doi.org/10.1017/fmp.2015.3 || For $E_7$, Freudenthal's paper ‘Sur le groupe exceptionnel $E_7$’, Nederl. Akad. Wetensch. Proc. Ser. A 56 = Indagationes Math. 15 (1953), 81–89. gives a quartic polynomial on the vector space underlying a the smallest nontrivial rep of $E_7$ (not clear whether complex or compact) whose automorphisms give the group. $\endgroup$ – David Roberts Jan 29 '19 at 23:41
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    $\begingroup$ I like this question but I agree with Anton Fetisov that it does not seem to be focused enough for MathOverflow. Probably the answer to the question as stated ("have mathematicians tried to invent...") is a simple "no." $\endgroup$ – Timothy Chow Jan 30 '19 at 3:22
  • $\begingroup$ @davidroberts I read some Freudenthal papers. It is not easy to follow. I believe, he constructed not compact version of the group - at least in "Beziehungen...". I am looking for more direct definition of the group as matrix group over quaternions. What geometry can be found behind the quartic form which you mention ? $\endgroup$ – user21230 Jan 30 '19 at 7:13

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