 Solution

The relation $[h,e]=2e$ forces $e{w}_i$ to have weight $a+2(i+1)$, sofor scalars $c_i$. The relation $[e,f]=h$ becomesWith $c_0=b$, this recurrence has the unique solutionfor every $i\in \mathbb{Z}$. Direct substitution also verifies $[h,f]=-2f$ and $[h,e]=2e$, so these formulas define the unique required sl2 Lie algebra action. They form an Intermediate-series sl2 module.