Lie group commutator bound $|[g,h]|< C|g||h|$

Question

So I'm reading a paper and I don't understand why this property holds. Namely, suppose we have a Lie Group $G$ and define a smooth norm on it, take $|g|=d(g,1)$ for example, then for sufficiently small neighborhood around $1$, any elements $g,h$ in that neighborhood satisfy $|[g,h]|< C|g||h|$ for some constant $C$. Can someone explain the details why this is true.