Given a finite Lie algebra that is generated by a set of rank-1 idempotents $\{P_{j}\}_{j}$ over the complex field, what are the conditions for it to be solvable (or alternatively nilpotent).
A constraint that feels natural to obtain this is to require that for any $P_{j}$, there are no more than $T$ different $P_{k}$ that don't commute with $P_{j}$. However you can construct counter examples when only this constraint is imposed.
Obviously if these generators all commute then the algebra is solvable, but similarly this is not the most general class of operators.