lie algebras - Meaning of vanishing Lie bracket

This is somewhat extension of question in why does Lie bracket of two coordinate vector fields always vanish?

Now i want to understand the meaning of vanishing Lie bracket.

$i.e$, For vector field $X$, $Y$ If

\begin{align}
[X,Y]=0
\end{align}
for all $Y$ on $M$,

Of course i know if $X, Y$ are coordinate basis, then $[X,Y]=0$, but here $Y$ can be arbitrary.

Borrow some logic from usual elementary algebra gives

$ax=0$ for all $x$ means $a=0$

Can i apply same thing here?

If $X=0$ then it obviously satisfied $[X,Y]=0$ for all $Y$ but i am uncomfortable with its inverse.