Any smooth vector field is a linear combination of left invariant vector fields

Question

Suppose that $G$ is a lie group and that $X$ is a smooth vector field on it. Is it true that $X$ is a linear combination of left invariant vector fields with smooth maps as coefficients?

Answer 1

Let $G$ be a Lie group and ${\cal G}$ its Lie algebra. Suppose that $e_1,...,e_n$ is a basis of ${\cal G}$, you can defined the left-invariant vector field $X_i(g)=(dL_g)_I(e_i)$, $(X_1(g),...,X_n(g))$ is a basis of the tangent space $T_gG$. Let $(a_1,...,a_n)$ be the dual basis of $(e_1,...,e_n)$. Let $\alpha^i$ be the $1$-form defined by $\alpha^i_g(u)=a_i(dL_g^{-1}(u))$. For every vector field $X$, $X(g)=\alpha^1_g(X)X_1+...+\alpha^n_g(X)X_n$.