TITLE: Beautiful descriptions of exceptional groups
QUESTION [51 upvotes]: I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need not so many words to describe this).
For $G_2$ we know the automorphisms of octonions and the rolling distribution (and also the intersection of three $Spin_7$-s in $Spin_8$).
For $F_4$ we know the automorphisms of Jordan algebra $H_3(\mathbb O)$ and Lie algebra of commutators of right multiplications in this algebra (see Chevalley-Schafer's paper for details).
For $E_6$ we know the automorphisms of determinant in $H_3(\mathbb O)$ and Lie algebra linearly spanned by right multiplications and $\mathfrak f_4$.
For $\mathfrak f_4$, $\mathfrak e_6$, $\mathfrak e_7$, $\mathfrak e_8$ we know the Vinberg-Freudenthal Magic Square.
What do we know (expressing in a simple form) about $E_7$ and $E_8$?

REPLY [10 votes]: Here is a description that is new and you can judge whether it is beautiful.  Given any simple complex Lie group $G$ and almost any irreducible representation $V$, the stabilizer of almost any $G$-invariant polynomial $f$ on $V$ has identity component $G$.
Cartan's examples 

$G = E_6$, $V$ of dimension 27, and $f$ cubic; or
$G = E_7$, $V$ of dimension 56, and $f$ quartic

are very special cases of this general principle.  (They are very special because in these cases the ring of $G$-invariant polynomials on $V$ is generated by $f$.)
In the case of the group $E_8$, you can take $V$ to be the Lie algebra $\mathfrak{e}_8$.  Then the ring of invariant polynomial functions is a polynomial ring with generators of degree 2 (the Killing quadratic form), 8, 12, 14, 18, 20, 24, 30.  The new result says: If you take $f$ to be any of the generators besides the Killing form, then $E_8$ is the identity component of the stabilizer of $f$.
This is a very concrete description of $E_8$, because an explicit formula for the degree 8 polynomial is already in the literature (Cederwall and Palmkvist - The octic $E_8$ invariant (arXiv)).
Alternatively, there is a commutative, nonassociative, and $E_8$-invariant product on its 3875-dimensional irreducible representation, and the automorphism group of this nonassociative ring is $E_8$.
There is also a variation on the result I mentioned at the beginning that may be worth mentioning: you can also realize each simple complex Lie group $G$, up to isogeny, as the stabilizer of a cubic form on some representation.  For $E_8$, you can take the cubic form to be the one defining the multiplication on the 3875-dimensional representation.
The new results mentioned here are from Garibaldi and Guralnick - Simple groups stabilizing polynomials (MSN, arXiv).