Every left invariant vector field on a Lie group is smooth. Spivak.

Question

Hello. I'm studying A comprehensive introduction to differential geometry by Spivak's and I'm stuck on the first two equalities.

Why $Xx^{i}(a)={L_a}_*X_e(x^i)=X_e(x^i\circ L_a)$?

Specifically, what complicates me is that $X$ is a vector field, then $X:G\to G_{e}$ but $(x,U)$ is a chart for $e$, i.e., $x(e)\in x(U)\subset\mathbb{R}^n$ open subset. Then, $x^{i}(a)\in \mathbb{R}$ but $X:G\to G_e$...How is the vector field $X$ interpreted to evaluate to $x^{i}(a)$ a real number?

Answer 1

There is a misunderstanding of notation here. $X x^i(a)$ is supposed to mean $(X x^i)(a)$.

The convention in differential geometry is usually that the action of a vector on a function takes precedence over function evaluation.