I know that $G\_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^\*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe in his answer to the post [A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$](https://mathoverflow.net/questions/252517/a-question-on-complex-semisimple-lie-groups-and-mathbbc2-omega) a nice description of the complex $E\_6$ as the group of symmetries of $V = \Lambda^2\_0 (\mathbb{C}^8)^\*$ endowed with a cubic form on that space. In that description, what is the real structure which gives the compact real form $E\_6$? I conjecture it would be induced by the real structure $j \wedge j$ on $V$, where $j$ is a quaternionic structure on $\mathbb{C}^8$ such that $\omega$ (see Prof. @RobertBryant's answer in the above link) is real. It is just a guess. Is it correct please?


I do not really know how to realize the compact real forms of $F\_4$, $E\_7$ and $E\_8$ as matrix groups. Your help is kindly appreciated. References are more than welcome (especially if they can be found online, and I hope "my" library has access to them!). If someone feels like writing a whole answer, then that would be great too. It is time for me to learn more about the exceptional Lie groups.


One last thing. I know that what I am looking for can be found in the relevant E. Cartan's papers. However, while I would definitely learn a lot by going back to the source, yet I don't have as much free time nowadays as I would like to (not to mention that reading Cartan is known to be difficult, and it is not the language barrier, in my case). So is there a simplified and modernized version of that part of Cartan's work please, that would also discuss compact real forms?



How to describe the compact real forms of the exceptional Lie groups as matrix groups?
Cartan describes all of the compact real forms of the simple Lie groups over $\mathbb{C}$ in his first paper that classifies the real forms. In fact, he describes them exactly in the terms that you ask for: A representation of the complex Lie group together with an auxilliary structure, either a real structure on the complex representation space or a Hermitian quadratic form.


For $\mathrm{G}\_2$ (resp, $\mathrm{F}\_4$, $\mathrm{E}\_8$), the compact forms are represented as special orthogonal real matrices of rank 7 (resp., 26, 248). For $\mathrm{E}\_6$ (resp. $\mathrm{E}\_7$), the compact real forms are represented as special unitary matrices of rank $27$ (resp. $56$).


Explicitly, here are the defining structures in the lowest dimensional representations of the compact real forms of the exceptional groups:


* $\mathrm{G}\_2$ is the stabilizer of a $3$-form on a real vector space of dimension $7$.
* $\mathrm{F}\_4$ is the stabilizer of a quadratic form and a cubic form on a real vector space of dimension $26$. (I believe that the cubic form alone is enough to define $\mathrm{F}\_4$.)
* $\mathrm{E}\_6$ is the stabilizer of a cubic form and a positive definite Hermitian form on a complex vector space of dimension $27$. (The cubic form alone only defines the complex $\mathrm{E}\_6$.)
* $\mathrm{E}\_7$ is the stabilizer of a symplectic form, a quartic form, and a positive definite Hermitian form on a complex vector space of dimension $56$. (The quartic form and the Hermitian form by themselves are almost enough to define $\mathrm{E}\_7$; they define a group with two connected components, the identity component of which is the compact $\mathrm{E}\_7$.)
* $\mathrm{E}\_8$ is the stabilizer of a $3$-form on a real vector space of dimension $248$.


