Extension of Dual Verma Module In Humphrey's BGG Category $\mathcal{O}$ Exercise 3.3 he asks:
Does the short exact sequence below always split?
$$ 0 \rightarrow M(\lambda) \rightarrow E \rightarrow M(\mu)^\check{} \ \rightarrow 0.$$
Having tried to show this it may possible to prove that it does always split if the map $M(\mu)^\check{} / L(\mu) \cong M(\nu)$ for $\nu$ some lower weight than $\mu$. However I feel the statement is not true in general - both the isomorphism and the splitting of the sequence.
Any help would be greatly appreciated, thanks. 
 A: Let the Lie algebra be $\mathfrak{g}:=\mathfrak{sl}_2(\mathbb{C})$ spanned by $\{h,x,y\}$ with $[x,y]=h$, $[h,x]=+2x$, and $[h,y]=-2y$.  The category $\mathcal{O}$ for $\mathfrak{g}$ is defined with respect to the Borel subalgebra $\mathfrak{b}:=\mathbb{C}h\oplus \mathbb{C}x$ and with the Cartan subalgebra $\mathfrak{h}:=\mathbb{C}h$.  Identify $\mathfrak{h}^*$ with $\mathbb{C}$ via $\lambda\mapsto \lambda(h)$ for every $\lambda\in\mathfrak{h}^*$.
Consider the Verma module $\mathfrak{M}(-2)$.  It is simple (i.e., $\mathfrak{M}(-2)\cong\mathfrak{L}(-2)\cong \mathfrak{M}^\vee(-2)$).  Tensoring $\mathfrak{M}\left(-1\right)$ (which is a projective module as $-1$ is a dominant weight) with the $2$-dimensional simple module $\mathfrak{L}\left(1\right)$ yields the projective cover $\mathfrak{P}(-2)$ of $\mathfrak{L}(-2)\cong\mathfrak{M}(-2)$.  That is, we have a short exact sequence
$$0\to \mathfrak{M}(0) \to \mathfrak{P}(-2) \to \mathfrak{M}(-2)\to 0\,,$$
which does not split.  Taking the duality of this exact sequence and noting that $\mathfrak{M}(-2)\cong \mathfrak{M}^\vee(-2)$, we obtain the non-splitting exact sequence
$$0\to \mathfrak{M}(-2) \to \mathfrak{I}(-2) \to \mathfrak{M}^\vee(0)\to 0\,,$$
where $\mathfrak{I}(-2)=\mathfrak{P}^\vee(-2)$ is the injective hull of $\mathfrak{M}(-2)$.
