If $G$ is a semi-simple Lie group and $g\in G$, then $G$ has a bi-invariant metric which is a Riemmanian metric. My question is:
with respect to this metric does the left-translation map \begin{equation} h \mapsto g\cdot h \end{equation} define an isometry on $G$?
(I apologize in advance if this question is trivial).
This is almost tautological. "Bi-invariant" means the metric is invariant under both left and right translation. The resulting distance function satisfies, in particular, $$d(gx,gy)=d(x,y)$$ Hence left translation is an isometry (as is right translation).