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 $E_8$?
Best Answer
It is not always clear what one means by 'the simplest description' of one of the exceptional Lie groups. In the examples you've given above, you quote descriptions of these groups as automorphisms of algebraic structures, and that's certainly a good way to do it, but that's not the only way, and one can argue that they are not the simplest in terms of a very natural criterion, which I'll now describe:
Say that you want to describe a subgroup $G\subset \text{GL}(V)$ where $V$ is a vector space (let's not worry too much about the ground field, but, if you like, take it to be $\mathbb{R}$ or $\mathbb{C}$ for the purposes of this discussion). One would like to be able to describe $G$ as the stabilizer of some element $\Phi\in\text{T}(V{\oplus}V^\ast)$, where $\mathsf{T}(W)$ is the tensor algebra of $W$. The tensor algebra $\mathsf{T}(V{\oplus}V^\ast)$ is reducible under $\text{GL}(V)$, of course, and, ideally, one would like to be able to chose a 'simple' defining $\Phi$, i.e., one that lies in some $\text{GL}(V)$-irreducible submodule $\mathsf{S}(V)\subset\mathsf{T}(V{\oplus}V^\ast)$.
Now, all of the classical groups are defined in this way, and, in some sense, these descriptions are as simple as possible. For example, if $V$ with $\dim V = 2m$ has a symplectic structure $\omega\in \Lambda^2(V^\ast)$, then the classical group $\text{Sp}(\omega)\subset\text{GL}(V)$ has codimension $m(2m{-}1)$ in $\text{GL}(V)$, which is exactly the dimension of the space $\Lambda^2(V^\ast)$. Thus, the condition of stabilizing $\omega$ provides exactly the number of equations one needs to cut out $\text{Sp}(\omega)$ in $\text{GL}(V)$. Similarly, the standard definitions of the other classical groups as subgroups of linear transformations that stabilize an element in a $\text{GL}(V)$-irreducible subspace of $\mathsf{T}(V{\oplus}V^\ast)$ are as 'efficient' as possible.
In another direction, if $V$ has the structure of an algebra, one can regard the multiplication as an element $\mu\in \text{Hom}\bigl(V\otimes V,V\bigr)= V^\ast\otimes V^\ast \otimes V$, and the automorphisms of the algebra $A = (V,\mu)$ are, by definition, the elements of $\text{GL}(V)$ whose extensions to $V^\ast\otimes V^\ast \otimes V$ fix the element $\mu$. Sometimes, if one knows that the multiplication is symmetric or skew-symmetric and/or traceless, one can regard $\mu$ as an element of a smaller vector space, such as $\Lambda^2(V^\ast)\otimes V$ or even the $\text{GL}(V)$-irreducible module $\bigl[\Lambda^2(V^\ast)\otimes V\bigr]_0$, i.e., the kernel of the natural contraction mapping $\Lambda^2(V^\ast)\otimes V\to V^\ast$.
This is the now-traditional definition of $G_2$, the simple Lie group of dimension $14$: One takes $V = \text{Im}\mathbb{O}\simeq \mathbb{R}^7$ and defines $G_2\subset \text{GL}(V)$ as the stabilizer of the vector cross-product $\mu\in \bigl[\Lambda^2(V^\ast)\otimes V\bigr]_0\simeq \mathbb{R}^{140}$. Note that the condition of stabilizing $\mu$ is essentially $140$ equations on elements of $\text{GL}(V)$ (which has dimension $49$), so this is many more equations than one would really need. (If you don't throw away the subspace defined by the identity element in $\mathbb{O}$, the excess of equations needed to define $G_2$ as a subgroup of $\text{GL}(\mathbb{O})$ is even greater.)
However, as was discovered by Engel and Reichel more than 100 years ago, one can define $G_2$ over $\mathbb{R}$ much more efficiently: Taking $V$ to have dimension $7$, there is an element $\phi\in \Lambda^3(V^\ast)$ such that $G_2$ is the stabilizer of $\phi$. In fact, since $G_2$ has codimension $35$ in $\text{GL}(V)$, which is exactly the dimension of $\Lambda^3(V^\ast)$, one sees that this definition of $G_2$ is the most efficient that it can possibly be. (Over $\mathbb{C}$, the stabilizer of the generic element of $\Lambda^3(V^\ast)$ turns out to be $G_2$ crossed with the cube roots of unity, so the identity component is still the right group, you just have to require in addition that it fix a volumme form on $V$, so that you wind up with $36$ equations to define the subgroup of codimension $35$.)
For the other exceptional groups, there are similarly more efficient descriptions than as automorphisms of algebras. Cartan himself described $F_4$, $E_6$, and $E_7$ in their representations of minimal dimsension as stabilizers of homogeneous polynomials (which he wrote down explicitly) on vector spaces of dimension $26$, $27$, and $56$ of degrees $3$, $3$, and $4$, respectively. There is no doubt that, in the case of $F_4$, this is much more efficient (in the above sense) than the traditional definition as automorphisms of the exceptional Jordan algebra. In the $E_6$ case, this is the standard definition. I think that, even in the $E_7$ case, it's better than the one provided by the 'magic square' construction.
In the case of $E_8\subset\text{GL}(248)$, it turns out that $E_8$ is the stabilizer of a certain element $\mu\in \Lambda^3\bigl((\mathbb{R}^{248})^\ast\bigr)$, which is essentially the Cartan $3$-form on on the Lie algebra of $E_8$. I have a feeling that this is the most 'efficient' description of $E_8$ there is (in the above sense).
This last remark is a special case of a more general phenomenon that seems to have been observed by many different people, but I don't know where it is explicitly written down in the literature: If $G$ is a simple Lie group of dimension bigger than $3$, then $G\subset\text{GL}({\frak{g}})$ is the identity component of the stabilizer of the Cartan $3$-form $\mu_{\frak{g}}\in\Lambda^3({\frak{g}}^\ast)$. Thus, you can recover the Lie algebra of $G$ from knowledge of its Cartan $3$-form alone.
On 'rolling distributions': You mentioned the description of $G_2$ in terms of 'rolling distributions', which is, of course, the very first description (1894), by Cartan and Engel (independently), of this group. They show that the Lie algebra of vector fields in dimension $5$ whose flows preserve the $2$-plane field defined by $$ dx_1 - x_2\ dx_0 = dx_2 - x_3\ dx_0 = dx_4 - {x_3}^2\ dx_0 = 0 $$ is a $14$-dimensional Lie algebra of type $G_2$. (If the coefficients are $\mathbb{R}$, this is the split $G_2$.) It is hard to imagine a simpler definition than this. However, I'm inclined not to regard it as all that 'simple', just because it's not so easy to get the defining equations from this and, moreover, the vector fields aren't complete. In order to get complete vector fields, you have to take this $5$-dimensional affine space as a chart on a $5$-dimensional compact manifold. (Cartan actually did this step in 1894, as well, but that would take a bit more description.) Since $G_2$ does not have any homogeneous spaces of dimension less than $5$, there is, in some sense, no 'simpler' way for $G_2$ to appear.
What doesn't seem to be often mentioned is that Cartan also described the other exceptional groups as automorphisms of plane fields in this way as well. For example, he shows that the Lie algebra of $F_4$ is realized as the vector fields whose flows preserve a certain 8-plane field in 15-dimensional space. There are corresponding descriptions of the other exceptional algebras as stabilizers of plane fields in other dimensions. K. Yamaguchi has classified these examples and, in each case, writing down explicit formulae turns out to be not difficult at all. Certainly, in each case, writing down the defining equations in this way takes less time and space than any of the algebraic methods known.
Further remark: Just so this won't seem too mysterious, let me describe how this goes in general: Let $G$ be a simple Lie group, and let $P\subset G$ be a parabolic subgroup. Let $M = G/P$. Then the action of $P$ on the tangent space of $M$ at $[e] = eP\in M$ will generally preserve a filtration $$ (0) = V_0 \subset V_1\subset V_2\subset \cdots \subset V_{k-1} \subset V_k = T_{[e]}M $$ such that each of the quotients $V_{i+1}/V_i$ is an irreducible representation of $P$. Corresponding to this will be a set of $G$-invariant plane fields $D_i\subset TM$ with the property that $D_i\bigl([e]\bigr) = V_i$. What Yamaguchi shows is that, in many cases (he determines the exact conditions, which I won't write down here), the group of diffeomorphisms of $M$ that preserve $D_1$ is $G$ or else has $G$ as its identity component.
What Cartan does is choose $P$ carefully so that the dimension of $G/P$ is minimal among those that satisfy these conditions to have a nontrivial $D_1$. He then takes a nilpotent subgroup $N\subset G$ such that $T_eG = T_eP \oplus T_eN$ and uses the natural immersion $N\to G/P$ to pull back the plane field $D_1$ to be a left-invariant plane field on $N$ that can be described very simply in terms of the multiplication in the nilpotent group $N$ (which is diffeomorphic to some $\mathbb{R}^n$). Then he verifies that the Lie algebra of vector fields on $N$ that preserve this left-invariant plane field is isomorphic to the Lie algebra of $G$. This plane field on $N$ is bracket generating, i.e., 'non-holonomic' in the classical terminology. This is why it gets called a 'rolling distribution' in some literature. In the case of the exceptional groups $G_2$ and $F_4$, the parabolic $P$ is of maximal dimension, but this is not so in the case of $E_6$, $E_7$, and $E_8$, if I remember correctly.