What are some examples of simple or semisimple non-split real Lie algebras? By non-split, I mean that no Cartan subalgebra $\mathfrak{h}$ is such that $\mathrm{ad}(X)$ is diagonalizable for each $X \in \mathfrak{h}$.
Is there a list somewhere? Or how can non-splitness be detected?
Simple Lie algebras over $\mathbb{R}$ are classified up to isomorphism by their Satake diagrams. The absolutely simple ones (i.e. those which are not scalar restrictions of complex ones -- these can easily be amended) are listed, for example, here: Onishchik, Vinberg p. 229 et seqq. (With the exception of the anisotropic = compact ones mentioned in the comments, for which by a classical result by E. Cartan there is exactly one for each type (i.e. irreducible root system); for example the $\mathfrak{su}(n+1)$ are the compact forms of type $A_n$).
In that classification, the split ones are exactly the ones where the Satake diagram is the classical Dynkin diagram, i.e. has only white nodes and no arrows. The others (plus, as mentioned, 1) the scalar restrictions of the simple complex ones and 2) the compact ones) are all the non-split simple ones you are looking for.
