One-parameter group of Lie groups

Question

Let $G$ be a connected Lie group and $\eta \in \mathfrak g$ such that the closure of the one-parameter group of transformations associated with $\eta$ is compact. Can we conclude that the adjoint representation: $\text{ad}_{\eta}:\mathfrak g \to \mathfrak g $ by $\text{ad}_{\eta}(\xi)=[\eta,\xi]$ is semisimple?

Answer 1

Let $K$ be the closure of this one-parameter group. Then $K$ is compact abelian, so any representation of $K$ is completely reducible, and all irreducible representations are 1-dimensional. Applying this to the adjoint representation shows that $\mathrm{ad}_\eta$ is semisimple.