  Let $A$ be a finite dimensional unital commutative associative algebra and
let $B$ be a finite dimensional vertex $A$-algebroid such that its Levi factor
is isomorphic to $sl_2$. Under suitable conditions, we construct an
indecomposable non-simple $\mathbb{N}$-graded vertex algebra $\overline{V_B}$
from the $\mathbb{N}$-graded vertex algebra $V_B$ associated with the vertex
$A$-algebroid $B$. We show that this indecomposable non-simple
$\mathbb{N}$-graded vertex algebra $\overline{V_B}$ is $C_2$-cofinite and has
only two irreducible modules.
