Left invariant metrics on a Lie group coming from Lie algebras

Question

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. If we have an inner product on $\mathfrak{g}$ then we can create a left invariant metric $d_G$ on $G$ by translations. On the other hand the map $v\mapsto exp(v)$ is locally diffeomorphism so we have the pushforward $d_\mathfrak{g}$ of the metric from the Lie algebra induced by the inner product. Can we say something about the relation between these two metrics in a small enough neighborhood around the origin? I think there should be some constant C (which depends on the neighborhood) such that $\frac{1}{C}d_G\leq d_\mathfrak{g} \leq C d_G$. Similarly, if $\mathfrak{g}=V_1\oplus \cdots \oplus V_k$ we can define the map $(v_1,...,v_n)\mapsto exp(v_1)\cdots exp(v_n)$ which is also locally diffeomorphism. Is the same true here?