Several results are presented on the integrability and extendability of Lie algebras composed of unbounded skew-symmetric operators with a common dense domain in a Hilbert space. Integrability of a Lie algebra \(\mathfrak {g}\) refers to the existence of an associated unitary representation \(\mathcal {U}\) of the corresponding simply connected Lie group, such that \(\mathcal {U}\) of the corresponding simply connected Lie group, such that \(\mathfrak {g}\) is the differential of \(\mathcal {U}\) . The findings extend previous integrability results in the literature and introduce new insights even for the case of a single operator. Applications include the discovery of a new invariant for certain Riemann surfaces.