Let $\mathfrak{g}=\mathfrak{gl}\_3$ over $\mathbb{C}$ with positive roots
\begin{equation\*}
\Phi\_+=\{\alpha\_1=(1,-1,0),\alpha\_2=(1,0,-1),\alpha\_3=(0,1,-1)\}.
\end{equation\*}
Consider the morphism
\begin{align}
M((-2,1,1)) \oplus M((-1,-1,2) &\xrightarrow\phi M((0,-1,1))\\ (v\_{(-2,1,1)},v\_{(-1,-1,2)}) &\mapsto(y\_{\alpha\_1}^2-(y\_{\alpha\_1}y\_{\alpha\_3}+2y\_{\alpha\_2}))v\_{(0,-1,1)}
\end{align}
with respect to $\mathfrak{g}$ inside the BGG category $\mathcal{O}$. Notice that $\phi$ is the sum of the embeddings  

\begin{align}
M((-2,1,1)) &\xrightarrow{g} M((0,-1,1))\,,\,v\_{(-2,1,1)} \mapsto y\_{\alpha\_1}^2 v\_{(0,-1,1)}, \\
M((-1,-1,2)&\xrightarrow{h} M((0,-1,1))\,,\,v\_{(-1,-1,2)} \mapsto -(y\_{\alpha\_1}y\_{\alpha\_3}+2y\_{\alpha\_2}))v\_{(0,-1,1)}. 
\end{align}


Here $M(\lambda)$ is the Verma module with the highest weight $\lambda$, $v\_\lambda \in M(\lambda)$ is the corresponding highest weight vector and $y\_{\alpha\_i}$ is the standard generator in the weight space $\mathfrak{g}\_{-\alpha\_i}$.


Then I'm struggling to compute $\ker(\phi)$, as we have relations inside $U(\mathfrak{g})$. What is a way to tackle this problem?



Computing kernel in the category $\mathcal{O}$
Here is a [less direct](https://mathoverflow.net/a/376625), but shorter, proof using some non-trivial machinery. Denote by $s$, $t$ the simple reflections, and $M\_w := M(w \cdot 0)$ where $w \in W$, and $\cdot$ is the "shifted" action $w \cdot \lambda := w(\lambda+\rho)-\rho$, where $\rho:=(1,0,-1)$, and analogously the simple modules $L\_w$.


By the construction in [1, 4.2] we have a chain complex (called *generalized BGG complex*):
$$0 \to M\_{sts} \to M\_{st} \oplus M\_{ts} \to M\_{t} \to L\_t \to 0, $$
where the non-trivial maps are up to scalar your $(e,f)$ and $\phi$. Since $L\_t$ is a Kostant module (as is any simple module in $\mathcal{O}$ in rank $2$; you can check this by calculating the Kazhdan–Lusztig polynomials and using the last sentence in [1, 3.4]), by [1, Theorem 4.3.] the above complex is exact. Your claim follows.


[1]. Brian D. Boe & Markus Hunziker (2009) [Kostant Modules in Blocks of Category $\mathcal O\_S$](https://arxiv.org/abs/math/0604336), Communications in Algebra, 37:1, 323-356.


