What I'm trying to show:
Let $Y$ be a vector field on a Lie group $G$. If $G$ is connected and $[X,Y]=0$ for all left invariant vector field $X$, then $Y$ is right invariant.
I thought I could prove it using only the relations between left and right invariant vector fields, but I failed. I realized I wasn't using the connectedness of $G$. I'm having difficulty in understanding what role plays the connectedness of $G$.
Best Answer
We must remember two things :
Two vector fields $X,Y$, commute iff there flows commute, iff $Y$ is invariant by the flow of $X$.
The flow of the left invariant vector field $X$ is the right translation by $\exp t X(e)$, where $X(e)$ is the value at $e$ of $X$.
From this, if $Y$ commute with evry left invariant vector field, it is invariant by the right translation by $g= \exp X $ for $X$ in the Lie algebra of $G$.
Now, if the group $G$ is connected, it is generated by elements of the form $\exp X$, and the result follows.