Q:

Nilpotency and 2 dimensional abelian lie subalgebras

I am wondering is it really necessary to use the adjoint version of Engel’s theorem to prove that if every 2 dimensional lie sub-algebra of a given Lie algebra L is abelian then L is nilpotent?
I wonder about that because you can write every 2 dimensional Lie algebra as the span of x and y in L and then by the assumption the bracket is zero for every two elements that I could choose.
What am I missing here?

A:

I think you are missing, that two arbitrary elements in $L$ will not span a subalgebra of $L$. For example, if $L$ is the Heisenberg Lie algebra with basis $(x,y,z)$ and Lie brackets determined by $[x,y]=z$, then the Lie bracket of each two elements spanning a subalgebra is zero, but not in general. $x$ and $y$ have a non-trivial Lie bracket and they do  not span a subalgebra of $L$ (because the product is not "closed").

