How do Jordan algebras help one understand representations of exceptional Lie algebras? For this question I'm happy to take the complex numbers as the base field.
I've been trying to learn a little bit about the exceptional Lie algebras and for a while they seemed inaccessible. I looked at McCrimmon's nice survey on Jordan algebras: http://projecteuclid.org/euclid.bams/1183540925 and it seems at least that one can use these "new" (in the sense that they are new to me) gadgets to get a foothold on the exceptionals by realizing them as automorphisms, or derivations, or whatever of basically the same guy $H_3({\bf O})$ (Hermitian $3 \times 3$ octonion valued matrices). It seems interesting but since I also have no intuition for these new objects, I'd like to ask the following question:
Are there any available written accounts of using "Jordan theory" to get working models for the representations of the exceptional Lie algebras, or at least their fundamental representations?
This is coming from the perspective of someone who likes explicit constructions and being able to write down bases, matrices, etc. if desired.
I can at least appreciate the case of $G_2$: using the octonions, we get its minimal representation, and the other fundamental is its adjoint, but we can also get all of its irreducibles by applying Schur functors to its minimal and mimicking Weyl's "tracefree tensor" construction (Huang, Zhu, "Weyl's construction and tensor product decomposition for $G_2$ http://www.jstor.org/pss/119028). Okay not quite Jordan algebraish, and I don't expect something like this to be written down for other types, but hopefully this gives an idea of what I'm looking for.
 A: There are descriptions for all the exceptional Lie algebras in terms of some sort of nonassociative algebra, except for $E_8$.  (For $E_8$, you can say that it is the derivations of itself as a Lie algebra, bu this is hardly informative.)  In each case, one describes the Lie algebra or group as derivations or automorphisms of some algebraic structure on the smallest-dimensional irreducible representation of the algebra, roughly speaking.


*

*Type $G_2$: automorphisms of the octonions, as you say

*Type $F_4$: automorphisms of an Albert algebra (27-dimensional exceptional Jordan algebra), see e.g. Chapter IX in The Book of Involutions.

*Type $E_6$: easiest is to think of this as the group of norm isometries for an Albert algebra, which is totally sufficient for the complex numbers.  


Up to here, the book "Octonions, Jordan aglebras, and Exceptional groups" by Springer and Veldkamp is a good reference.


*

*Type $E_7$: Stabilizer of a quartic form and skew-symmetric bilinear form (making a "Freudenthal triple system" - this phrase will get you to a list of references) on its 56-dimensional irreducible representation.


My paper Structurable algebras and groups of type E6 and E7 gives a survey on a kind of nonassociative algebra that links types E6 and E7 in the same way that types F4 and E6 are linked (smaller one is an automorphism group, larger one is the isometries of a norm form).  You can see this from a different view in Springer's paper Some groups of type E7.  Or from the view of representation theory in Helenius's paper Freudenthal triple systems by root system methods.
Now a disclaimer.  You asked your question for the complex numbers, but I think this Jordan-theoretic interpretation does not have much value there because you already have the root space decomposition of the Lie algebra, etc.  Rather, the purpose of this viewpoint is to handle more general base fields and especially to handle anisotropic (over the real numbers, one usually says "compact") groups over those fields.  That's the whole idea behind the Book of Involutions: to use similar interpretations to study classical groups over general fields.
A: No, you cannot use "Jordan theory" to get working models for the representations of the exceptional Lie algebras. There is a chance of doing it with $F_4$ but not with any $E_n$-s. 
Historically, people used Jordan algebras to prove existence of exceptional Lie algebras but there are better ways nowadays such as Serre's relations or Freudental's magic squares...
Let me add that Jordan algebra is just a (2,1)-tensor preserved by the group. A more natural question is what tensors the group is a stabilizer of. 
