Sorry if this is too simple, but I'm having some trouble working through an exercise given to us by our professor. The exercise goes as follows:
For a Lie algebra $\frak{g}$ of dimension three, an element $ \alpha \in \frak g^*$ is called a contact form if $\alpha \wedge d\alpha \neq 0 $.
Show that for $\frak g^*$$ = (0,0,12)$, the form $\alpha = e_3$ is a contact form. Determine all the contact forms on this Lie algebra. Determine which of the other three-dimensional Lie algebras admit a contact form.
I have managed to show that $e_3$ is a contact form for $\frak g^*$$ = (0,0,12)$. I know that we have $$ de_1=0, \ \ de_2 = 0, \ \ de_3=e_1\wedge e_2 $$ and this gives $e_3 \wedge de_3 = e_3 \wedge e_1 \wedge e_2 \neq 0$, since ${e_1, e_2, e_3}$ are basis elements and thus linearly independent. However, I'm having trouble proceeding further than this and I would appreciate any advice or reference recommendations to solve this.