I am reading Thurston's book (Three dimensional geometry and topology), and there is something I can't understand.
Let $G$ be a Lie Group, in the book it says that there exists an $\epsilon > 0$ and $C > 0$ such that, if $d(1, a) < \epsilon$ and $d(1, b) < \epsilon$ then $$d(1, [a,b]) < Cd(1,a)d(1,b).$$
it doesn't say the metric but I am assuming Haar.
According to the book, this happens because of Taylor's theorem applied somehow to the map $[,]: G \times G \to G$, but I cannot understand how or to what apply Taylor's theorem.