Cartan subalgebra and Casimir invariants

To my understanding, if group $G$ is semisimple, $mathfrak{g}$ is its Lie algebra, and $Delta_T=[T,cdot]$ is the adjoint representation, one can analyze its spectrum with $[T,U]=lambda U$.
A particularly important value is $R=inf_{Tinmathfrak{g}}deg(lambda=0)$, since there are (at most) $R$ commuting generators of $mathfrak{g}$ and $R$ Casimir invariants. Now,

• the {$C_1,…,C_R$} commuting generators are the generators of a subalgebra called Cartan’s subalgebra, which is interesting because its structure constants are zero: $[C_i,C_j]=0 spaceforall space i,j$;
• the {$H_1,…,H_R$} Casimir invariants are defined by the fact that they commute with all the generators of $mathfrak{g}$.

My question: are $C_i$ and $H_i$ the same? Maybe under some hypotheses? It seems at least plausible, since they both are $R$ independent operators, and they both have banal commutators.