In Zee's book Group Theory in a Nutshell for Physicists, Chapter VI.3, it is stated that (quoted contents are in italic):
Consider a general Lie algebra with $n$ generators defined by
$$ [X^a, X^b] = i {f^{ab}}_c X^c \tag{1} $$
A footnote says that here we only discuss the algebras of compact Lie groups. On the next page, Zee argues that the structure constants should be real:
Hermiticity of $X^a$ implies that the ${f^{ab}}_c$ are real.
A few pages later, after converting to the Cartan-Weyl basis $H^i, E_\beta$ for which
$$ [H^i, E_\beta] = \beta^i E_\beta \tag{16} $$
Zee again assumes that $H^i$ are Hermitian to show that if $\beta$ is a root vector, so is $-\beta$. So my questions are:
Can the generators $\{X^a\}$ of a Lie algebra, or the $\{H^i\}$ in the Cartan-Weyl basis (or the representation matrices of them) always be made Hermitian? Are there any restrictions (e.g. compact, semi-simple, connected etc) on the Lie group to make this true? Please kindly provide references to related theorems.
(I am a physics student, and I apologize if my statements are not clear or rigorous enough.)