Let $G$ be a connected Lie group and $\mathfrak{g}$ its Lie algebra. Let $\exp:\mathfrak{g}\rightarrow G$ be the exponential map. A straightforward use of the Lie product formula shows that $exp$ induces a continuous surjective homomorphism of abelian groups from $(\mathfrak{g},+)$ onto $G/[G,G]$, the abelianization of $G$. Moreover, since $\exp\big[[\mathfrak{g},\mathfrak{g}]\big]\subseteq [G,G]$, we actually get that $\exp$ induces a continuous surjective homomorphism from $\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$ onto $G/[G,G]$, where $[\mathfrak{g},\mathfrak{g}]=\mathrm{span}_\mathbb{R}\{[X,Y]\colon X,Y\in\mathfrak{g}\}$.
So to summarize, we have obtained a short exact sequence $$0\to H\to \mathfrak{g}/[\mathfrak{g},\mathfrak{g}]\to G/[G,G]\to 0.$$ Can we abstractly describe what $H$ (the kernel of the above surjective homomorphism) is? I care only about matrix Lie groups if it makes any difference.