What is an example of a vector field that is not left-invariant?

Question

Let $G$ be a Lie group, $L_g$ the left-translation on this group with differential $d L_g$. A vector field $X$ on $G$ is called left-invariant if

$$ X \circ L_g = d L_g \circ X \quad \forall g \in G$$

i.e.

$$ X_{gh} = (d L_g)_h (X_h) \quad \forall g,h \in G. $$

Now, this definition seems so natural to me that I cannot come up with a non-trivial counterexample for a vector field that is $\textit{not}$ left-invariant. In my mind, pushing forward on the tangent space is basically always the same as the group action...

Could you provide me with such a counterexample that helps understand the notion of left-invariance?

Answer 1

You could start taking the Lie group $(\mathbb R^n,+),$ and considering what does it means for a vector field $X$ on $\mathbb R^n$ to be left-invariant.