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.

Topology of Lie groups, I and II, AMS, Translations of Mathematical Monographs, volume 91, 1991." also deals with exceptional Lie groups in part II, where the cohomology (mod p) is computed. $\endgroup$ – Oldřich Spáčil Dec 4 '12 at 15:13