Verma self-extensions in O split

Given: The setting above and the hypotheses in the statement.

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. The category $\mathcal{O}$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0\to A\to E\to B\to 0$ is exact, $A,B\in \mathcal{O}$, and $E$ is $\mathfrak{h}$-semisimple, then $E\in \mathcal{O}$. The middle-term weight hypothesis is essential. (Category O is abelian and extension closed among weight modules)

For a $\mathfrak{g}$-module $V$, sending a homomorphism $T:M(\lambda )\to V$ to $T(v_{\lambda })$ is a bijection onto the vectors $v\in V$ of weight $\lambda$ annihilated by ${\mathfrak{n}}^{+}$. Here $M(\lambda )$ is def-verma-module. The nonzero vectors in this target are precisely the highest-weight vectors of weight $\lambda$ from def-highest-weight-vector-and-cyclic-highest-weight-module; the zero vector corresponds to the zero homomorphism. (The universal property of Verma modules)

The weights of $M(\lambda )$ are exactly $\lambda -\beta$ for $\beta \in Q^{+}$; every weight space is finite dimensional, and $M(\lambda {)}_{\lambda }=\mathbb{C}{v}_{\lambda }$. (Weights of a Verma module lie below lambda)