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In this paper, a holomorphic vertex operator algebra $U$ of central charge 24with 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 withweight 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 vertexoperator algebra is also established. By a similar method, we also establishedthe uniqueness of a holomorphic vertex operator algebra of central charge 24with the weight one Lie algebra $E_{7,3}A_{5,1}$. As a consequence, we verifythat all $71$ Lie algebras in Schellekens' list can be realized as the weightone Lie algebras of some holomorphic vertex operator algebras of central charge$24$. In addition, we establish the uniqueness of three holomorphic vertexoperator algebras of central charge $24$ whose weight one Lie algebras have thetype $A_{8,3}A_{2,1}^2$, $F_{4,6}A_{2,2}$, and $E_{7,3}A_{5,1}$.

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