Question about lie bracket..

Question

Let $G$ be a Lie group with Lie algebras $\mathfrak{g}$ and let $\mathfrak{h}\subseteq \mathfrak{g}$ be a Lie subalgebra. Write $F_p=DR_p(e)\mathfrak{h}$, $p\in G$, where $R_p:G\rightarrow G$ given by $a\mapsto ap$ is the right translation. Suppose $X, Y\in \mathfrak{X}(G)$ are smooth fields and $$X(p), Y(p)\in F_p,$$ for all $p\in G$. With this hypothesis I can conclude $[X(e), Y(e)]\in \mathfrak{h}=F_e$. Does this imply $[X(p), Y(p)]\in F_p$ for all $p\in G$?

Answer 1

If you know $$[f_*X\ f_*Y]=f_*{[X\ Y]}$$ here $f_*$ denote $f$ tangent map. general speaking,$f$ is diffeomorphism. then this question is obvious. $$X(p)=DR_p(e)X_1\ X_1\in h$$ $$Y(p)=DR_p(e)X_1\ X_2\in h$$ \begin{eqnarray*} % \nonumber to remove numbering (before each equation) [X(p)\ Y(p)] &=& [DR_p X_1\ DR_p X_2 ]\\ &=& DR_p [X_1\ X_2 ]\in DR_p(h)=F_p \ \forall p\in G \end{eqnarray*}