I am trying to understand the proof of Kempf-Ness theorem, which relates polystability (Mumford's definition) and zeroes of moment maps. A key part of this proof is the following idea, that for a complex reductive group $G$ (By Weyl's result, this is the complexification of a compact lie group $K$), $G/K$ is geodesically complete and its geodesics are of the form $[e^{it\eta} g]$ where $\eta \in \mathfrak{k}$ and $g \in G$, essentially one parameter subgroups.
Why is this true? I do know that $K$ is compact and hence it has a bi-invariant metric coming from the Killing form. Can I somehow extend this metric to $G$ and get a bi-invariant metric on $G$? I do know that for bi-invariant metrics, one will have the connection given by $[X,Y]$ and this would give rise to the required geodesics. Any help here will be appreciated.