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 $mathfrak{g}=so(2n), ngeq 3$ and $k$ is an admissible number for $hat{mathfrak{g}}$. It is also shown that the diagonal coset vertex operator algebra $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)otimes L_{sl_2}(4,0))$ is rational and $C_2$-cofinite in case $k$ is an admissible number for $hat{sl_2}$. Furthermore, irreducible modules of $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)otimes L_{sl_2}(4,0))$ are classified in case $k$ is a positive odd integer.