Let $g$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let ${u_j}$ and ${u^j}$ be bases of $g$ dual with respect to a nondegenerate invariant bilinear form $(cdot|cdot)$ on $g$, and consistent with the triangular decomposition of $g$. Let $L(Lambda)$ be an integrable representation of $g$ with highest weight $Lambda$, and let $v_Lambda$ be its highest weight vector. Denote the Casimir $Omega=sum_j u_jotimes u^j$.

I want to know why $Omega(v_Lambdaotimes v_Lambda)=(Lambda|Lambda)v_Lambdaotimes v_Lambda$? Could someone give some explanation or some references?