My question is about the construction of Verma module of a lie algebra $L$, there is one step in the construction which I do not quite understand.
Let $L=N_-\oplus H\oplus N_+$ be the triangular decomposition of a lie algebra $L$ over the field $\mathbb{C}$. Let $U(L)$ be the universal enveloping algebra of $L$. Define $I_\lambda$ be the left idea of $U(L)$ generated by $N_+$ and $h-\lambda(h)1$:
$$I_\lambda=\{xn_++y(h-\lambda(h)1)\ |\ x,y\in U(L),n_+\in N_+,h\in H\}.$$
I want to show that $I(\lambda)\cap N_-=(0)$. This is easy at the first sight, but then I found it's subtle to use the PBW theorem directly: when assuming $x,y$ are standard PBW basis monomials, and substitute them into the expressions above, one does not get a standard linear combinations of standard monomials. Can anyone help me fix this problem?