Background/motivation: One of the usual constructions of [the adjoint representation of] the $E_8$ exceptional Lie group (found, e.g., in J. F. Adams's, "Lectures on Exceptional Lie Groups", esp. chap. 6–7) consists of starting from $\mathit{Spin}(16)$ and taking the direct sum of the latter's adjoint representation and one half-spin representation (it remains, of course, to construct a Lie bracket on this direct sum, but at least three of the four cases to consider are clear). The reason for this approach is that this direct sum is the branching of the adjoint representation of $E_8$ (to be constructed) to its maximal subgroup $D_8 = \mathit{Spin}(16)$. The branching $E_8 \to D_8$ is arguably the most manageable one, so it makes sense to use it to construct $D_8$. However, branching to the $A_8$ subgroup is also intelligible, so I ask:

Question: Given that the adjoint representation of $E_8$, restricted to its maximal subgroup $A_8$ (meaning $\mathit{SL}(9)$ or $\mathit{SU}(9)$ according as we are working with complex or real compact Lie groups), decomposes as $\bigwedge^3 V \oplus \bigwedge^3 V^* \oplus W$ where $V$ is the ($9$-dimensional) natural representation of $A_8$, $V^*$ is its dual, and $W$ is the ($80$-dimensional) adjoint representation of $A_8$ (the nontrivial factor in $V\otimes V^*$), is there an explicit description of a Lie bracket on $\bigwedge^3 V \oplus \bigwedge^3 V^* \oplus W$ that constructs $E_8$? Is this description somewhere to be found in the literature?

(To be honest, what I am really interested in is how to get an intuitive grasp of this branching. But I presume the easiest way to do that is to use it to construct $E_8$, rather than hope to describe it starting from some other construction of $E_8$. However, any such description counts as an answer to this question.)


An excellent reference for this kind of descriptions (and for many other facts about $E_8$!) is Skip Garibaldi's paper "$E_8$, the most exceptional algebraic group" in the Bulletin of the AMS (here). In particular, in section 4, he describes various possible branchings, and Example 4.4 deals with the branching $A_8 \subset E_8$.

The specific example is rather short, but he includes some further references that might be helpful:

  • Hans Freudenthal, "Sur le groupe exceptionnel $E_8$", Nederl. Akad. Wetensch. Proc. Ser. A 56 (=Indag. Math. 15) (1953), 95–98 (=pages 284–287 in his Selecta published by the EMS in 2009).

  • William Fulton & Joe Harris, Representation Theory: A First Course (Springer 1991, GTM 129), exercise 22.21 on page 361.

  • John Faulkner, "Some forms of exceptional Lie algebras", Comm. Algebra 42 (2014), 4854–4873 (=arXiv:1305.0746).

[Freudenthal's paper is completely explicit and elementary; Fulton & Harris provide a bit more theoretical background; Faulkner is more general and works over an arbitrary commutative ring. —Gro-Tsen]

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    $\begingroup$ Great! If you don't mind, for the sake of MO's completeness, I'll edit your answer to link to the three references given by Garibaldi (which, indeed, answer my question fairly well). $\endgroup$ – Gro-Tsen Jun 13 '17 at 15:03
  • $\begingroup$ Of course I don't mind :) $\endgroup$ – Tom De Medts Jun 13 '17 at 15:08

Of course, the original description of this branching is due to Élie Cartan, himself. For example, see Chapitre IX of his 1914 paper Les groupes réels simples, finis et continu (Annales scientifiques de l'É.N.S. 3rd serie, 31 (1914), 263–355).

His chapter on $\mathrm{E}_8$ starts on page 328, and it is very clear and straightforward. The formulae he gives, both for the complex and real forms, are exactly what you are looking for. I think it's the best reference for this, myself. It's available for free download from Nundam, but I don't have the link in front of me right now.

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