Let $G$ and $H$ be Lie groups equipped with left invariant metrics, and let $F \colon G \to H$ be an identity-preserving isometry.
Does the differential $dF$ commute with left translations? More precisely, if $v \in T_{e}G$ ($e$ being the identity element of $G$), is it true that
$$ \left(L_{F(p)}\right)_{\ast} \left(dF_{e}(v) \right) = dF_{p}\left( \left(L_{p}\right)_{\ast}(v) \right) \,? $$
EDIT: An equivalent question would be: Is the image of a left invariant vector field under $dF$ also left invariant? I believe this is not true in general, but maybe under some extra assumptions on $G$ or $H$?