Reference request: Calculation in exceptional Lie groups

Question

Let $G$ be a compact connected simple exceptional Lie group. Let $G$ be contained in a unitary group ${\rm U}(n)$ by some standard (low dimensional) unitary representation. For example in the case of $E_8$ one could take the adjoint representation acting on the $e_8$-Lie algebra. Then there is a maximal torus $T$ of $G$ that is contained in the maximal torus of diagonal matrices in ${\rm U}(n)$ and a set of simple roots corresponding to $T$.

I am looking for a reference, where the characters of $T$ corresponding to simple roots are computed explicitly.

For an instance in the case of the classical group ${\rm SU}(n)$ the $i$-th character maps the diagonal element ${\rm diag}(t_1,\ldots, t_n)$ to $t_it_{i+1}^{-1}$ and this can hardly be made more explicit. The only book dealing with the exceptional groups beyond their classification I found so far is "Lectures on exceptional Lie groups" by J.F. Adams, but the point I am interested in is not taken into account therein.

Answer 1

But, can't you read this off the table of simple roots for each of the exceptional Lie algebras?

For example, $\mathrm{G}_2$ contains $\mathrm{SU}(3)$ as a subgroup and the maximal torus of $\mathrm{SU}(3)$ is also a maximal torus of $G_2$. The table of simple roots then implies, in the notation you established above, that the two characters on this maximal torus corresponding to the simple roots are $t_1{t_2}^{-1}$ and ${t_1}^{-1}{t_2}^{2}{t_3}^{-1}$. Then to know how this torus acts on the weight spaces of any representation of $\mathrm{G}_2$, you just need to know the weights in the representation.

For all of the so-called 'fundamental representations' of the exceptional groups (of which there are $r$ for each group of rank $r$), these were worked out by Cartan almost a hundred years ago and explicitly listed in his paper classifying the representations of the exceptional groups.

See É. Cartan, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. S. M. F. t. 41 (1913), p 53–96. (available to be dowloaded on Nundam)