  In this paper, under the assumption that the diagonal coset vertex operator
algebra $C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak
g}(l,0))$ is rational and $C_2$-cofinite, the global dimension of
$C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak g}(l,0))$ is
obtained, the quantum dimensions of multiplicity spaces viewed as
$C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes L_{\mathfrak
g}(l,0))$-modules are also obtained. As an application, a method to classify
irreducible modules of $C(L_{\mathfrak g}(k+l,0),L_{\mathfrak g}(k,0)\otimes
L_{\mathfrak g}(l,0))$ is provided. As an example, we prove that the diagonal
coset vertex operator algebra $C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes
L_{E_8}(2,0))$ is rational, $C_2$-cofinite, and classify irreducible modules of
$C(L_{E_8}(k+2,0),L_{E_8}(k,0)\otimes L_{E_8}(2,0))$.
