First of all, I am so sorry if this question is not appropriate to be here. I tried to ask something similar on Math Stack Exchange but it didn't have much attention. Any comment and I delete the question.


I was reading the classical paper from Milnor entitled Curvature of Left Invariant Metrics on Lie Groups. The approach is to consider an orthonormal frame on the Lie algebra, since all geometric information is gained considering an inner product on it vector space, once we have the correspondence between left invariant metrics and inner products on the Lie algebra. 


From this is easy to take information about Levi-Civita connection, Curvatures and etc. But my question is, what about the geodesics? Is there any formula that relates the frame on the Lie algebra and the geodesic equation? Is possible to describe all the geodesics just looking for the orthonormal frame on the Lie algebra?


If it is not the case, what is the general approach to obtain geodesics on Lie groups? The only way is trying to describe the general Levi-Civita connection? 



Geodesics equation on Lie groups with left invariant metrics
Let $\nabla$ be the connection induced by the Levi-Civita connection. $\nabla$ is left invariant. It thus defines a bilinear product $b$ on ${\cal G}$ the Lie algebra of $G$. Let $c(t)$ be a geodesic. We can write $\dot c(t)=dL\_{c(t)}(x(t))$ where $x(t)\in {\cal G}$. It you write the equation $\nabla\_{\dot c(t)}\dot c(t)=0$, you obtain 


$\dot x(t)+b(x(t),x(t))=0$.


