From a few examples I can see why the defintion is the way it is, but, I would like a better understanding of why the dimension of the Cartan subalgebra is $2n-\text{rank}(A)$ where $A$ is the Cartan matrix.

To be specific why does the Lie algebra associated to the universal extention $\tilde{L}SU(2)$ not have an associated Cartan matrix while the affine Lie algebra associated to $\mathbb{T}\ltimes\tilde{L}SU(2)$ does?