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 $(\mathbb{C}^2, \omega)$ 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?