# Left multiplication isometry?

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 $$h \mapsto g\cdot h$$ 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).