This is the definition of cartan subalgebra define in Brian Hall, Lie groups, Lie algebras and representations, 2nd Edition, Chpt 7, Sec 2, Def. 7.10.
I am assuming the ground field is $C$ or it does not make sense to talk about diagonalizability.
If $g$ is complex semisimple lie algebra(here he assumed semisimple=reductive+triviality of center), and $h\subset g$ is an ideal s.t. $(1)\forall H_i\in h$, $[H_i,H_j]=0$
$(2)\forall G\in g, [G,h]=0\implies G\in h$
$(3)$ For all $H\in h, ad_H$ is diagonalizable.(i.e. adjoint representation of $h$ is diagonalizable.)
It is clear that $(1)$ and $(2)$ are required to find maximal amount of joint eigenvalues of $ad_H$. However, since $[H_i,H_j]=0$ and $ad:g\to gl(g)$ is lie algebra homomorphism, certainly it suffices to demand only 1 particular $H$ s.t. $ad_H$ diagonalizable.
$\textbf{Q:}$ Do I always need diagonalizable condition? What is the counter example that I do need? Since there is requirement for lie algebra homomorphism for $ad$, there will be constraint on the structure of $ad$. Furthermore, if I do need diagonalizability, can I just use one per reasoning above?