A Lie algebra with two of its Lie bracket are zero while the third is not. Is it possible to construct a system of three vector field u,v and w an $R^{3}$ such that [u,v]=0=[u,w], but [v,w]$\neq$0?
I tried to solve it by applying the Jacobi identity property which makes the 2nd and 3rd term vanish so I am left with [u[v,w]]=0. I am stuck here. Any help please. Thanks in advance

Yes, consider the 2 dimensional non commutative Lie algebra, $[a,b]=a$, add $c$, in the center $[c,a]=[c,b]=0$
