Consider the three dimensional Lie algebra with basis $x,y,z$ and commutator relations $$[x,y]=y-z$$ $$[y,z]=0$$ $$[z,x]=y-z$$ I want to find a faithful matrix representation of this Lie algebra, by Ado's theorem one exists. The adjoint representation is not faithful so it needs to be some other set of matrices which give this.
I have tried to find a three $3\times3$ matrices which satisfy the above by brute force but the complexity and runtime of such a programme is extremely high. If anyone knows a solution or a practical way of finding one that would be great.