A unitary vertex operator algebra arising from the 3C-algebra

We give an algebraic proof of the unitarity of the vertex operator algebra $L(21\mathrm{/}22,0)\oplus L(21\mathrm{/}22,8)$ and of all its irreducible ordinary modules, using a coset realization arising from the $3C$-algebra. Motivated by the structure of the resulting module decomposition, we establish a general result on fusion rules for commutant vertex operator subalgebras within the framework of modular tensor categories. As an application of this general result, we explicitly determine the fusion rules of all irreducible $L(21\mathrm{/}22,0)\oplus L(21\mathrm{/}22,8)$-modules.