EDIT: I think the partial answer to my question is that in order to talk about weights we always need a complex Lie algebra. If the Lie algebra is real, we use the complexification, This is necessary, because otherwise the Chevaller/Cartan Weyl basis does not exist and we don't have roots that act as ladder operators etc. All finite-dimensional semisimple complex Lie algebras have a Chevaller-basis.
Nevertheless, it would be great if someone with more experience could help me understand these things a bit more clearly
In the appendix A page 20 of this paper the authors write
...complex weight [ $| (1 / 2 , \sqrt{3/2}) \rangle$], which cannot be accomplished with a single hermitian matrix
What do the authors mean by this? Why is a weight of the adjoint representation automatically complex and why doesn't correspond it to a singlet hermitian matrix?
I thought the adjoint representation is always real and thus the weights should correspond to real matrices.
On the next page they refer to the combination $| (1 / 2 , \sqrt{3/2}) \rangle + | (-1 / 2 , -\sqrt{3/2}) \rangle $ as "real combination".
Why do we need to add two weights of the adjoint representation in order to get sth real?
As an example take $\mathfrak{su}(3)$. We write the adjoint $8$ as a $3 \times 3$ matrix because $3 \otimes \bar 3= 1 \oplus 8$. Then components are given through the tensor products $ 8_{ij}= 3_i \otimes \bar 3_j$, where we add the $i$-th weight vector of $3$ to the $j$-th weight vector of $\bar 3$. This yields the matrix $$ 3 \otimes 3= \left( \begin{array}{ccc} \left(\begin{array}{cc} 0 & 0 \\\end{array}\right) & \left( \begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\\end{array}\right) & \left(\begin{array}{cc} 1 & 0 \\ \end{array} \right) \\ \left(\begin{array}{cc} -\frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \end{array}\right) & \left( \begin{array}{cc}0 & 0 \\ \end{array} \right) & \left( \begin{array}{cc} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \end{array} \right) \\ \left( \begin{array}{cc} -1 & 0 \\ \end{array} \right) & \left( \begin{array}{cc} -\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \end{array} \right) & \left( \begin{array}{cc} 0 & 0 \\ \end{array} \right) \\ \end{array} \right)$$
This tells us, for example, that the weight $( \begin{array}{cc} \frac{1}{2} & \frac{\sqrt{3}}{2} \\\end{array} )$ corresponds to the matrix
$$\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right) $$
which I would call a real matrix.