As in the title I looking for technics to compute weights of kernels, images or cokernels in category $\mathcal{O}$ besides checking everything directly by hand.
To be more concrete, consider the following example coming from the strong BGG resolution of $L((0,0,0))$ (based on Morphism of Verma modules)
\begin{equation} M((-2,1,1)) \oplus M((-1,-1,2)) \stackrel{\delta_{2_1}}{\rightarrow} M((0,-1,1)) \end{equation}
where we have for respectively maximal vectors $v_{(-2,1,1)}$, $v_{(-1,-1,2)}$ and $v_{(0,-1,1)}$ that
\begin{equation*} \delta_{2_1}(v_{(-2,1,1)},v_{(-1,-1,2)})= (b_1y_{\alpha_1}^2+b_2(y_{\alpha_1}y_{\alpha_3}+2y_{\alpha_2}))v_{(0,-1,1)} \end{equation*} with $b_i \in \{-1,1\}$.
As we have relations in $U(\mathfrak{n}^-)$, I already struggle to compute for example the kernel directly and in general it will get even harder. So I was wondering if there are other ways to get the weights of the kernel etc.?
