$\newcommand\su{\operatorname{su}}$I am trying to find the branching rules for the coset construction: $$\widehat{\su}(4)_2 \rightarrow \frac{\widehat{\su}(4)_2}{\widehat{\su}(2)_4} \oplus \widehat{\su}(2)_4,$$ where the subscript indicates the level of the affine Lie algebra, and the coset model is $\frac{\widehat{\su}(4)_2}{\widehat{\su}(2)_4}$. If it is helpful, the central charges are: $$c_{\widehat{su}(4)_2} = 5,\quad c_{\widehat{\su}(2)_4} = 2, \quad\text{and}\quad c_{\frac{\widehat{\su}(4)_2}{\widehat{\su}(2)_4}} = 3.$$
If anyone knows the branching rules, or knows a procedure by which this can be computed, it would be much appreciated. (As an example of something I am looking for, is Table I in https://arxiv.org/pdf/hep-th/9309093.pdf, where the branching rules for the maverick coset $\frac{\widehat{su}(3)_2}{\widehat{su}(2)_8}$ are listed.)
