Consider a Lie algebra $\mathcal{G}$ of the form $$[T_i, T_j] = f_{ij}^k T_k$$ which has an Abelian (corresponding to the Abelian Lie algebra $A$) central extension $\mathcal{H}$ of the form $$[T'_i, T'_j]_{\mathcal{H}} = [T_i, T_j]_{\mathcal{G}}+\Theta(T_i, T_j)$$ such that $\Theta: \mathcal{G} ~\times \mathcal{G} \to A$ and $T'_i \in \mathcal{H}$.
I wish to know if I construct a haar measure on $\mathcal{G}$ can that help me find a haar measure on $\mathcal{H}$?