How do spinors arise in systems of less than 3 effective spatial dimensions as representations of the Lorentz group?

There is a systematic way to define spinor representations in any number of dimensions. The key idea is the following. The SO$(d-1,1)$ algebra can be written

$$i[ Sigma^{munu}, Sigma^{sigmarho}]=eta^{nusigma} Sigma^{murho} + eta^{murho}Sigma^{nusigma} - eta^{nurho} Sigma^{musigma} -eta^{musigma} Sigma^{nurho}$$ where $eta^{munu}=$diag(-1,+1,...,+1) and $Sigma^{munu}$ is an antisymmetric matrix which packages the rotation and boost generators $J_i,K_i$. For instance in $d=3+1$ we have $$Sigma^{munu}=begin{pmatrix}0 &K_1 & K_2 & K_3-K_1 & 0 & J_3 & -J_2-K_2 & -J_3 & 0 & J_1 -K_3 & J_2 & -J_1 & 0end{pmatrix}. $$

Moreover, whenever we have $d$ Dirac matrices $Gamma^{mu}$ satisfying

$${ Gamma^mu,Gamma^nu} = 2eta^{munu}$$

we automatically have a representation of SO($d-1,1$) using

$$Sigma^{munu} = -frac{i}{4} [Gamma^mu,Gamma^nu]$$ as one can check. So all we need are suitable $Gamma$s and we are done; the spinors can be identified by building raising/ lowering operators out of the $Gamma$s as usual. In $d=1+1$ we can use

$$Gamma^0 = begin{pmatrix} 0 & 1 -1 & 0end{pmatrix}quadGamma^1 = begin{pmatrix} 0 & 1 1 & 0end{pmatrix}$$ and in dimension 2+1 we can tack on $Gamma^2 = Gamma^0Gamma^1$. There's a systematic way to get higher dimensions, too, and to find out which representations are irreducible. (Note the number of components of the spinor goes like $2^{d/2}$ for $d$ even or $2^{(d-1)/2}$ for $d$ odd).