Exponents of a semisimple Lie algebra I'd like to compute the exponents of a semisimple complex Lie algebra $\mathfrak{g}$. According to http://math.berkeley.edu/~theojf/LieQuantumGroups.pdf proposition 8.1.2.18, this amounts to decomposing the adjoint representation $\mathrm{ad} : \mathfrak{sl}_2(\mathbb{C}) \rightarrow \mathrm{End}(\mathfrak{g})$ of the principal $\mathfrak{sl}_2(\mathbb{C})$ into irreps.
It's written "This is a little bit of work, but you know all the weights, so you know how to do it". I understand this means we should know the character of this representation. But how do you know it? 
Thanks
 A: The computation of exponents can be done by reducing to the simple summands of the Lie algebra.
If $\mathfrak{g}$ is a finite dimensional complex semisimple Lie algebra, you can decompose it into its simple summands
$$\mathfrak{g} = \bigoplus_{i=1}^{n} \mathfrak{g}_{i}.$$
Then, if $\{H, X, Y\}$ is the principal $\mathfrak{sl}_{2}$ triple, you can show that 
$$H = \bigoplus_{i=1}^{n} H_{1}, X = \bigoplus_{i=1}^{n} X_{1}, Y = \bigoplus_{i=1}^{n} Y_{1}$$
where $\{H_{i}, X_{i}, Y_{i}$ form the principal $\mathfrak{sl}_{2}$ triple for $\mathfrak{g}_{1}.$
This implies that if $\mathfrak{g}_{i} = V_{p_{i_{1}}} \oplus \cdots \oplus V_{p_{i_{m_{i}}}}$ as an irreducible decomposition in terms of $\{H_{i}, X_{i},Y_{i}\}$ modules, then the irreducible decomposition of $\mathfrak{g}$ in terms of principle $\mathfrak{sl}_{2}$ modules is
$$\oplus_{i = 1}^{n} \left(V_{p_{i_{1}}} \oplus \cdots \oplus V_{p_{i_{m_{i}}}}\right).$$ 
This shows, by the Proposition you cited, that the exponents of $\mathfrak{g}$ are the disjoint unions of the exponents of the $\mathfrak{g}_{i}$.
Now, calculating exponents of the simple $\mathfrak{g}_{i}$ can either be done by hand (because the weights are the roots, and decomposing $\mathfrak{sl}_{2}$ modules is quite easy), or you can just look it up. Of course, in high enough dimension, this could be a little difficult to compute.
EDIT: Also, to decompose $\mathfrak{g}$ into simple summands, you just have to compute its Dynkin diagram or Cartan matrix and look at each connected component.
