Problem: Prove that (up to isomorphisms) exists a unique Lie algebra $\mathfrak{g}$ with $\dim(\mathfrak{g}) = 3$ such that $\dim([\mathfrak{g},\mathfrak{g }]) = 1$ and $[\mathfrak{g},\mathfrak{g}]\subseteq Z(\mathfrak{g})$.
I don't even know how to start, I guess that is not too difficult but I need an initial guide.
Some pointers to get you started.
As a vector space $\langle z \rangle = [\mathfrak{g}, \mathfrak{g}]$ for some $z \in \mathfrak{g}$. Note that $z \in Z(\mathfrak{g})$ by assumption.
Extend to a basis $z,x,y$ of $\mathfrak{g}$. What can you say about the commutators between the basis elements?