Integrability criterion for abelian extensions of Lie groups

We establish a criterion for when an abelian extension of infinite-dimensional Lie algebras $\mathfrak {\hat {g}} = \mathfrak {g} \oplus _\omega \mathfrak {a}$ integrates to a corresponding Lie group extension $A \hookrightarrow \widehat {G} \twoheadrightarrow G$, where $G$ is connected, simply connected and $A \cong \mathfrak {a} / \Gamma$ for some discrete subgroup $\Gamma \subseteq \mathfrak {a}$. When $\pi _1(G)\neq 0$, the kernel $A$ is replaced by a central extension $\widehat {A}$ of $\pi _1(G)$ by $A$.