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.

allfundamental representations. $\endgroup$ – Jim Humphreys Sep 27 '10 at 18:17