Construct representation of Lie algebra

Question

Consider the 3D Lie algebra with the following defined:

$ [x,y] = z $, $ [x,z]=0 $, $ [y,z]=0 $

From this, I want to get a little help how to start finding a 3D representation by $3\times 3$ real matrices. I don't know how to begin.

Answer 1

Since the Lie algebra is 3-dimensional, you can consider the adjoint representation.

If we write $L$ your Lie algebra, then for each $x\in L$ there is a linear map $$\operatorname{ad}(x):y\in L\mapsto [x,y]\in L,$$ and this defines a map $$\operatorname{ad}:L\to\operatorname{End}(L)$$ which you can easily check to be a representation.

(Yoy could also consider the trivial representation, but that is slightly less entertaining...)