Rationality and $C_2$-cofiniteness of certain diagonal coset vertex operator algebras

In this paper, it is shown that the diagonal coset vertex operator algebra $C(L_{\mathfrak{g}}(k+2,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(2,0))$ is rational and $C_2$-cofinite in case $\mathfrak{g}=so(2n),n\ge 3$ and $k$ is an admissible number for $\widehat{\mathfrak{g}}$. It is also shown that the diagonal coset vertex operator algebra $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$isrationaland$C_2$-cofiniteincase$k$ is an admissible number for $\widehat{s{l}_2}$. Furthermore, irreducible modules of $C(L_{s{l}_2}(k+4,0),L_{s{l}_2}(k,0)\otimes L_{s{l}_2}(4,0))$ are classified in case $k$ is a positive odd integer.