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
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 $\mathfrak{g}_{i}.$
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 $$\bigoplus_{i = 1}^{n} \left(V_{p_{i_{1}}} \oplus \cdots \oplus V_{p_{i_{m_{i}}}}\right).$$
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.