The Bianchi type IV Lie algebra (do they call it L(3,3)? ), $$ [y,z] = 0, \qquad [x,y] = y, \qquad [x, z] = y + z , $$ has the evident adjoint representation by 3×3 matrices.
Would it further have a 2-dimensional rep, 2×2 matrices—analogous to Pauli's for su(2)?
No. Indeed, after conjugation, the image would be exactly the Lie algebra of upper triangular matrices, which is not isomorphic to yours since it has a nontrivial center.
Actually, over the complex numbers every Lie subalgebra of $\mathfrak{gl}_2$ is isomorphic to $\{0\}$, $\mathfrak{a}$ the 1-dimensional abelian Lie algebra, $\mathfrak{b}$ (the 2-dim non-abelian Lie algebra), $\mathfrak{sl}_2$, or the direct product of one of these three by $\mathfrak{a}$.