In [1] they say that the Cartan subalgebra of a loop algebra $\mathring{g} = \mathbb C[t,t^{-1}]\otimes g$ is $\mathring h = \mathbb C[t,t^{-1}]\otimes h$, where $h$ is the Cartan subalgebra of the simple Lie algebra $g$.
On the other hand, in [2] they say that the Cartan subalgebra of $\mathring g$ is $1\otimes h$ instead.
How can this be?
My guess:
Physicists actually care about the untwisted affine Kac Moody algebra $\hat g = \mathring g \oplus \mathbb C K \oplus \mathbb C \delta$, which has Cartan subalgebra generated by $\{1\otimes H^1,\ldots, 1\otimes H^r,K,\delta\}$, where the $H^i$ span $h$. In fact, adding the derivatoin $\delta$ has the effect of "shrinking" the Cartan subalgebra, which otherwise would be spanned by $\{t^n\otimes H^i,K\}_{i=1,...,r}^{n\in \mathbb{Z}}$. This is what I understood form DiFrancesco's book on CFT.
So Kerf seems to cut the middle man and focus directly on the part of the Cartan subalgebra which interests the physicists. I am not sure about this, though.
[1] Senesi, Prasad. "Finite-dimensional representation theory of loop algebras: a survey." Quantum affine algebras, extended affine Lie algebras, and their applications 506 (2010): 263-283.
[2] De Kerf, Eddy A., Gerard GA Bäuerle, and A. P. E. Ten Kroode. Lie algebras, Part 2: Finite and infinite dimensional Lie algebras and applications in physics. Vol. 7. Elsevier, 1997.