In their original paper, Kazama and Suzuki treated the coset models $G/H$ such that this quotient is a special kind of Kahler manifold called hermitian symmetric space. They give a bunch of examples, including Grassmanian manifolds $\frac{SU(m+n)}{SU(m)\times SU(n)\times U(1)}$.

However, in another paper (by Behr and Fredenhagen) they say that the Kazama-Suzuki models are of the form $\frac{G_k\times SO(2d)_1}{H}$, and gave the example of the Grassmanian manifold, which they wrote as $\frac{SU(m+n)_1\times SO(2nk)_1}{SU(m)_{k+1}\times SU(n)_{n+1}\times U(1)}$. The subscripts just refer to the level of the representation of the loop algebras that they want to take when constructing the coset using the GKO method.

Why is it OK to add the $SO(2d)$? It should affect the coset, and we get a different representation of the $N=2$ super-CFT if we add it. How can these two definitions be compatible?