Extensions of complex tori by copies of $\mathbb{C}^{\times}$

Question

Let $$1 \rightarrow (\mathbb{C}^{\times})^r \rightarrow G \rightarrow V/L \rightarrow 0$$

be an extension of complex lie groups, where $L$ is a lattice in a complex vector space $V$.

Is $G$ necessarily commutative?

Answer 1

$G$ acts on $(\mathbb{C}^\times)^r$ by conjugation, but $(\mathbb{C}^\times)^r$ is commutative so the action factors through the quotient. However the automorphism group of $(\mathbb{C}^\times)^r$ is discrete (it's $GL_r(\mathbb{Z})$) so the action of $V/L$ must be trivial and therefore the group is commutative.