Question about calculating Lie bracket given a three dimensional Lie algebra

Question

Suppose we have $\frak{g}\in\mathbb{R^3}$ spanned by $X, Y, Z$ such that $[X,Y]=Y, [X,Z]=Y+Z$. What is $[Y, Z]$?

I tried to expand the bracket, $[X, Y]=XY-YX=Y, [Y, X]=YX-XY$, but don't see how to proceed otherwise.

Answer 1

Unfortunately, I'm not very familiar with Lie brackets, so I only have a partial answer via the properties from their wikipedia page. Hopefully someone else can do the final step.

Recall the Jacobi Identity $$[X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0$$ As $[X,Y]=Y$, this gives us that $$[X,[Y,Z]]+[Z,Y]+[Y,[Z,X]]=0$$ Also, recall that the Lie bracket is antisymmetric, so $[X,Y]=-[Y,X]$. We can further reduce this to $$[X,[Y,Z]]-[Y,Z]+[Y,-[X,Z]]=[X,[Y,Z]]-[Y,Z]+[Y,-Y-Z]=0$$ Finally, the Lie bracket is bilinear, so we can expand the last bracket, getting: $$[X,[Y,Z]]-[Y,Z]-[Y,Y]-[Y,Z]=0$$

$[\cdot,\cdot]$ being antisymmetric implies that $[Y,Y]=0$, so we have that $$[X,[Y,Z]]-2[Y,Z]=0$$ Hopefully, from here the answer is much more achievable.