I have the following question, motivated by the expression for the character of level 1 highest weight integrable representations of simply-laced affine algebras (in terms of the string function). It follows from the character expression (by comparing the leading "conformal dimension") that for a fundamental weight $\omega$ with mark 1 the following identity should hold in simply-laced case: $$ \frac{(\omega,\omega+2\rho)}{2(\mathfrak{g}^\vee+1)}=\frac{1}{2}|\omega|^2=\frac{1}{2}(\omega,\omega), $$ where $\mathfrak{g}^\vee$ is the dual coxeter number of the simply-laced simple $\mathfrak{g}$. The lhs is the quadratic Casimir divided by $2(\mathfrak{g}^\vee+1)$. I checked it for $A_n$ series and $E_6,\,E_7$ explicitly, but I feel there should be some simple general argument.

So the question is if there indeed is a simple argument why this should be true? Is there any generalization?