The sl2 Verma embedding chain

Since $\rho =1$ and $s\cdot m=-m-2$, the relevant pairing is $m+1$, which is positive integral exactly when $m\in {\mathbb{Z}}_{\ge 0}$. The PBW vectors $f^r{v}_m$ ($r\ge 0$) form a basis of $M(m)$. The relations in the Given imply $h{f}^r{v}_m=(m-2r)f^r{v}_m$ and, by induction from $e{f}^{r+1}{v}_m=fe{f}^r{v}_m+h{f}^r{v}_m$, imply $e{f}^r{v}_m=r(m-r+1)f^{r-1}{v}_m$ for $r\ge 1$.

For $m\in {\mathbb{Z}}_{\ge 0}$, step 1.1 makes the nonzero PBW vector $f^{m+1}{v}_m$ singular of weight $m-2(m+1)=-m-2$. The Verma universal property gives a nonzero map $M(-m-2)\to M(m)$, and the cited injectivity lemma makes it the displayed embedding, including $m=0$. For any $m\in \mathbb{C}$, the PBW weights of $M(m)$ are $m-2r$ for integers $r\ge 0$. A nonzero map $M(s\cdot m)\to M(m)$ must send its highest vector to a nonzero vector of weight $s\cdot m=-m-2$, so $r=m+1\in {\mathbb{Z}}_{\ge 0}$. If $r>0$, then $m\in {\mathbb{Z}}_{\ge 0}$; if $r=0$, then $m=-1$ and $s\cdot m=m$, so the map is a scalar identity on the cyclic module rather than a nontrivial reflection embedding. No other $m$ permits the claimed direction.