**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.)



Constructing $E_8$ from its branching to $A_8$
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)](http://www.ams.org/journals/bull/2016-53-04/S0273-0979-2016-01540-0/).
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](http://www.tandfonline.com/doi/abs/10.1080/00927872.2013.822878)", *Comm. Algebra* **42** (2014), 4854–4873 (=[arXiv:1305.0746](https://arxiv.org/abs/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](https://mathoverflow.net/users/17064/gro-tsen)]


