Questions about non-abelian 2 dimensional lie algebras

Question

I have some troubles understanding the non-abelian 2 dimensional lie algebras. And the question might be trivial.

For a non-abelian 2 dimensional lie algebras $L$, Then, $S = span\{x,y\}$and $[x,y]=x$. My question is that: can I say $[y,x]=y$?

Since $L$ is lie algebra, I have $[x,y]=-[y,x]$, would this imply $x=-y?$ (if x=-y,then x and y are not linearly independent...would this imply L is 1-dim?)

I am kind of lost in this...can anyone help me please!

Answer 1

What you can say is, that all $2$-dimensional non-abelian Lie algebras, over any field K are isomorphic, so that we can describe it by the brackets $[x,y]=x$ for a basis $(x,y)$, as well as by the brackets $[y,x]=y$. In general, all non-abelian Lie algebras $L$ of dimension $2$ satisfy $\dim [L,L]=1$, so that we can write $[x,y]=rx+sy$ for some $r,s\in K$, not both zero.

Answer 2

No, you can't say that. Take for example the sublagebra of $\mathfrak{sl}_2$ spanned by $$e=\begin{bmatrix}0&1\\0&0\end{bmatrix}\;\;\;\mbox{and}\;\;\;h=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$$ then $[h,e]=2e=-[e,h]$. Now, take $x=e$ and $y=\frac12h$.