I am interested in the details of Elie Cartan's thesis, and, more specifically the explicit construction of the exceptional Lie groups as groups of symmetries of some specific homogeneous polynomials (according to what I have read in many places). I am interested in the details. For instance, what does one such a polynomial look like? What do all such polynomials (with groups of symmetry a fixed exceptional Lie group) look like? Is the set of all such polynomials dense in the relevant polynomial space? 


Also, I am interested in different realizations of the exceptional Lie groups: for instance, $G\_2$ is the group of symmetries of a skew-symmetric trilinear form on $\mathbb{C}^7$. And in this case, the space of such skew-symmetric trilinear forms is dense in the set of all skew-symmetric trilinear forms on $\mathbb{C}^7$.


So I guess in general, I am interested in concrete and explicit realizations of the exceptional Lie groups as groups of symmetries of a specific algebraic object. Also what is the set of all such objects with this property? And, is the set of all such objects dense in the relevant vector space? I hope my questions are sufficiently clear!



Where can I find details of Elie Cartan's thesis?
$\mathrm{G}\_2$ is the only one of the exceptional groups that can be defined as the stabilizer of a `generic' tensorial object on a vector space and, over the complex numbers, even this is not quite right.


More precisely, let $V$ be a vector space (over $\mathbb{F}$, which could be $\mathbb{R}$ or $\mathbb{C}$) of dimension $d$ and let $\mathrm{Sch}(V)$ be some Schur functor applied to $V$. For example, $\mathrm{Sch}(V)$ might be $S^k(V^\*)$, the symmetric homogeneous polynomials of degree $k$, or $\Lambda^k(V)$, the exterior $k$-forms of degree $k$. Or it might be something less standard, such as the kernel of the natural map
$$
W:\Lambda^k(V)\otimes V \longrightarrow \Lambda^{k+1}(V)
$$
or the natural trace map
$$
\mathrm{tr}: V\otimes V^\* \to \mathbb{F}.
$$


It is very rare that $\mathrm{GL}(V)$ acts on $\mathrm{Sch}(V)$ with open orbits outside of the classical cases, such as 
$$
\mathrm{Sch}(V) = V,\ V^\*,\ S^2(V),\ S^2(V^\*), \Lambda^2(V), \Lambda^2(V^\*)
$$
or these tensored with some power of the $1$-dimensional representations $\Lambda^d(V)$ or $\Lambda^d(V^\*)$. When $d=2$, you also have $S^3(V)$ and $S^3(V^\*)$, and when $d=6,7,8$, you have $\Lambda^3(V)$ and $\Lambda^3(V^\*)$ (and, of course, these tensored with some power of the $1$-dimensional representations $\Lambda^d(V)$ or $\Lambda^d(V^\*)$). That's about it. 


When $\mathrm{GL}(V)$ acts on $\mathrm{Sch}(V)$ with an open orbit $\mathcal{O}\subset\mathrm{Sch}(V)$, we say that an element $w\in\mathcal{O}$ is *stable*. The $\mathrm{GL}(V)$-stabilizer of such a stable element $w$ is the group $G\_w\subset\mathrm{GL}(V)$. 


In the classical cases, these stabilizers are essentially the orthogonal and symplectic groups. In the case $d=7$ and $\mathrm{Sch}(V)=\Lambda^3(V^\*)$ or $\Lambda^3(V)$ (or these twisted by some power of $\Lambda^7(V)$ or $\Lambda^7(V^\*)$), the identity component of the stabilizer of a stable element is $\mathrm{G}\_2$. These are the *only* cases in which the stabilizer of a stable element is (up to finite extension) an exceptional group.


The other exceptional groups $G\subset \mathrm{GL}(V)$ do occur as stabilizers of *non-stable* elements of some $\mathrm{Sch}(V)$. For example, when $d=26$, there is an element of $S^3(V^\*)$ whose stabilizer has identity component $\mathrm{F}\_4$, when $d=27$, there is an element of $S^3(V^\*)$ whose stabilizer has identity component $\mathrm{E}\_6$, when $d=56$ there is an element of $S^4(V^\*)$ whose stabilizer has identity component $\mathrm{E}\_7$, and when $d=248$ there is an element of $\Lambda^3(V^\*)$ whose stabilizer has identity component $\mathrm{E}\_8$. None of these are stable elements, though.


(These are the answers when $\mathrm{F}=\mathrm{C}$. The story in the case of real forms is more complicated. For that, you should look in Cartan's 1913 paper classifying the real forms of the complex simple groups.)


