Given a Dynkin diagram, we can construct a presentation of a semisimple Lie algebra $\mathfrak g$ by the Cartan elements $H_\alpha$, raising operators $E_i$ and lowering operators $F_i$ satisfying the Serre relations. These are analogous to the ladder operators encountered in physics. However, there is a prominent feature of ladder operators not present a priori: Ladder operators work on Hilbert spaces (let's restrict to finite dimensional spaces for simplicity), and there is a certain adjoint relation between the operators.
Take $\mathfrak{so}(3)$ for example. We can generate this algebra by $L_z, L_+$ and $L_-$. Here $L_z$ is hermitian, serving as the Cartan subalgebra; $L_+$ and $L_-$ are the ladder operators, satisfying $L_+^\dagger = L_-$. This prompts the question:
Is it possible to equip all the finite dimensional complex representations of semisimple Lie algebras with an inner product, so that $H_\alpha$ is (anti)hermitian, and $E_i^\dagger = F_i$?
Notice that this is different from unitary representations for Lie groups (or, after a differentiation, a "hermitian representation" for Lie algebras) since not everything is hermitian. It also probably depends on the specific choice of Cartan algebra.
