Definition- vector field corresponding to a point

Question

Let $G$ be a Lie group, and let $x\in Lie(G)$, what is the definition of the vector field on $G$ corresponding to x?

I'm reading Alexander Kirillov's book and he mentions this object many times, and I can't find any formal definition to the "corresponding to $x$".

Thanks in advance.

Answer 1

I suppose that Kirillov defines the Lie algebra $\mathfrak g$ of $G$ as the space of all left-invariant vector fields on $G$ (a standard definition). Even if he doesn't define $\mathfrak g$ this way, it is natural that he proves that there is a natural bijection from $\mathfrak g$ onto that space. Therefore, each element of $\mathfrak g$ is a vector field on $G$ or, at least, corresponds to such a vector field.