How can one infer from the Lie's theorem (in terms of existence of a common eigenvector) that a complex irreducible representation of a solvable lie algebra has dimension 1?
What I know is that one can write $\sigma(x)(v) \in \mathbb{C}.v$ if $\sigma$ is the representation so that $\mathbb{C}.v$ is invariant. How to complete the argument from here?