A Holomorphic vertex operator algebra of central charge 24 with weight one Lie algebra $F_{4,6}{A}_{2,2}$

In this paper, a holomorphic vertex operator algebra $U$ of central charge 24 with the weight one Lie algebra $A_{8,3}{A}_{2,1}^2$ is proved to be unique. Moreover, a holomorphic vertex operator algebra of central charge 24 with weight one Lie algebra $F_{4,6}{A}_{2,2}$ is obtained by applying a ${\mathbb{Z}}_2$-orbifold construction to $U$. The uniqueness of such a vertex operator algebra is also established. By a similar method, we also established the uniqueness of a holomorphic vertex operator algebra of central charge 24 with the weight one Lie algebra $E_{7,3}{A}_{5,1}$. As a consequence, we verify that all $71$ Lie algebras in Schellekens’ list can be realized as the weight one Lie algebras of some holomorphic vertex operator algebras of central charge $24$. In addition, we establish the uniqueness of three holomorphic vertex operator algebras of central charge $24$ whose weight one Lie algebras have the type $A_{8,3}{A}_{2,1}^2$, $F_{4,6}{A}_{2,2}$, and $E_{7,3}{A}_{5,1}$.