I have a rather simple question. Let $$ 1 \to T^k \to G \to T^l \to 1 $$ be a central extension of Lie groups. Is it true that $G \cong T^{k+l}$? If not what can it be and what are some simple hypotheses that guarantee that it is a torus?
I am having a hard time finding references on extensions of Lie groups beyond papers written in the 1940s. The fact that they are quite scattered makes it difficult for me to learn even the basics on this topic, so I would also love to have recommendations on references (possibly textbooks).