Verma module associated to a Lie algebra

Question

Let $L$ be a Lie algebra with Cartan decomposition $L=H \oplus(\sum_{\alpha \in \Phi}L_{\alpha})$. Let $B=\sum_{\alpha \in \Phi^+}L_{\alpha}$ and $N=\sum_{\alpha \in \Phi^-}L_{\alpha}$ so that $L=B \oplus N$. By definition Verma module is $\Delta(\lambda)=\mathcal{U}(L)\otimes_{\mathcal{U}(B)}\mathbb{C}_{\lambda}$ where $\mathcal{U}(L)$ is the universal enveloping algebra and $\mathbb{C}_{\lambda}$ is a one dimensional $\mathcal{U}(B)$ module. Now we can regard $\Delta(\lambda)$ as a $\mathcal{U}(N)$ module also and as $\mathcal{U}(N)$ modules I was able to prove that $\Delta(\lambda)$ is isomorphic to $\mathcal{U}(N) \otimes _{\mathbb{C}}\mathbb{C}_{\lambda}$.From this I want to conclude that $\Delta(\lambda)$ is a $\mathcal{U}(N)$ free module of rank 1. Also the map from $\mathcal{U}(N) \to \Delta(\lambda)$ mapping $n \to n.v_{\lambda }$ is an isomorphism where $v_{\lambda}$ is the canonical generator of $\Delta(\lambda)=\mathcal{U}(L)\otimes_{\mathcal{U}(B)}\mathbb{C}_{\lambda}$.