  The problem of determining maximal ideals in universal affine vertex algebras
is difficult for levels beyond admissible, since there are no simple character
formulas which can be applied. Here we investigate when certain quotient
$\mathcal V$ of universal affine vertex algebra $V^k(\mathfrak{g})$ is simple.
We present a new method for proving simplicity of quotients of universal affine
vertex algebras in the case of affine vertex algebra
$L_{k_n}(\mathfrak{sl}_{2n})$ at level $k_n:=-\frac{2n+1}{2}$. In that way we
describe the maximal ideal in $V^{k_n}(\mathfrak{sl}_{2n})$. For that purpose,
we use the representation theory of minimal affine $W$-algebra
$W^{min}_{k_{n+1}}(\mathfrak{sl}_{2n+2})$ developed in [2]. In particular, we
use the embedding $L_{k_n}(\mathfrak{sl}_{2n}) \subset
W^{min}_{k_{n+1}}(\mathfrak{sl}_{2n+2})$ and fusion rules for
$L_{k_n}(\mathfrak{sl}_{2n})$--modules. We apply this result in the cases
$n=3,4$ and prove that a maximal ideal is generated by one singular vector of
conformal weight $4$. As a byproduct, we classify irreducible modules in the
category $\mathcal{O}$ for the simple affine vertex algebra
$L_{-7/2}(\mathfrak{sl}_{6})$.
