From the relativistic covariance of the Dirac equation (see Section 2.1.3 in the QFT book of Itzykson and Zuber for a derivation. I also more or less follow their notation.), you know how a Dirac spinor transforms. One has $$\psi'(x')=S(\Lambda)\ \psi(x)$$
under the Lorentz transformation
$$x'^\mu= {\Lambda^\mu}_\nu\ x^\nu= {\exp(\lambda)^\mu}_\nu\ x^\nu=(I + {\lambda^\mu}_\nu+\cdots)\ x^\nu\ .$$
Explicitly, one has $S(\Lambda)=\exp\left(\tfrac{1}8[\gamma_\mu,\gamma_\nu]\ \lambda^{\mu\nu}\right)$.
To show reducibility, all you need is to find a basis for the gamma matrices (as well as Dirac spinors) such that $[\gamma_\mu,\gamma_\nu]$ is block diagonal with two $2\times 2$ blocks. Once this is shown, it proves the reducibility of Dirac spinors under Lorentz transformations since $S(\Lambda)$ is also block-diagonal. Such a basis is called the chiral basis. It is also important to note that a mass term in the Dirac term mixes the Weyl spinors in the Dirac equation but that is not an issue for reducibility.
While this derivation does not directly use representation theory of the Lorentz group, it does use the Lorentz covariance of the Dirac equation. I don't know if this is what you wanted.
(I am not interested in your bounty -- please don't award me anything.)
Recall a Dirac spinor which obeys the Dirac Lagrangian
$$\mathcal{L} = \bar{\psi}(i\gamma^{\mu}\partial_\mu -m)\psi.$$
The Dirac spinor is a four-component spinor, but may be decomposed into a pair of two-component spinors, i.e. we propose
$$\psi = \left( \begin{array}{c} u_+\\ u_-\end{array}\right),$$
and the Dirac Lagrangian becomes,
$$\mathcal{L} = iu_{-}^{\dagger}\sigma^{\mu}\partial_{\mu}u_{-} + iu_{+}^{\dagger}\bar{\sigma}^{\mu}\partial_{\mu}u_{+} -m(u^{\dagger}_{+}u_{-} + u_{-}^{\dagger}u_{+})$$
where $\sigma^{\mu} = (\mathbb{1},\sigma^{i})$ and $\bar{\sigma}^{\mu} = (\mathbb{1},-\sigma^{i})$ where $\sigma^{i}$ are the Pauli matrices and $i=1,..,3.$ The two-component spinors $u_{+}$ and $u_{-}$ are called Weyl or chiral spinors. In the limit $m\to 0$, a fermion can be described by a single Weyl spinor, satisfying e.g.
$$i\bar{\sigma}^{\mu}\partial_{\mu}u_{+}=0.$$
Majorana fermions are similar to Weyl fermions; they also have two-components. But they must satisfy a reality condition and they must be invariant under charge conjugation. When you expand a Majorana fermion, the Fourier coefficients (or operators upon canonical quantization) are real. In other words, a Majorana fermion $\psi_{M}$ may be written in terms of Weyl spinors as,
$$\psi_M = \left( \begin{array}{c} u_+\\ -i \sigma^2u^\ast_+\end{array}\right).$$
Majorana spinors are used frequently in supersymmetric theories. In the Wess-Zumino model - the simplest SUSY model - a supermultiplet is constructed from a complex scalar, auxiliary pseudo-scalar field, and Majorana spinor precisely because it has two degrees of freedom unlike a Dirac spinor. The action of the theory is simply,
$$S \sim - \int d^4x \left( \frac{1}{2}\partial^\mu \phi^{\ast}\partial_\mu \phi + i \psi^{\dagger}\bar{\sigma}^\mu \partial_\mu \psi + |F|^2 \right)$$
where $F$ is the auxiliary field, whose equations of motion set $F=0$ but is necessary on grounds of consistency due to the degrees of freedom off-shell and on-shell.
Best Answer
Perhaps the shortest answer (from my elementary understanding of representation theory) would be that the differences between the types of spinors you asked for lie primarily in terms of the representations of the rotation group under which they transform: If the spinors are symbolised by ψ, then the transformation rule:
will have:
M = SL(2,C) for 2-component Weyl (and other relativistic, e.g. Lorentz) spinors, which obey the Weyl equation (the massless form of the Dirac equation)
M = SU(2) for the non-relativistic 2-component (Pauli) spinors, which obey the Schrodinger-Pauli equation – the non-relativistic but massive limit of the Dirac equation (I suspect their components, for normalised wave functions are each restricted to the unit circle in the complex plane.)
M = something much more general than either of the above for Cartan spinors in their most general form (possibly via SO(p,q) for general p, q?) as Cartan claimed his spinors are the most general mathematical form of spinors, and they deal with rotations in spaces of any number of dimensions. They should therefore range beyond 2-component objects (as do even Weyl spinors for e.g. 6-D space).
Although you profess disinterest in Dirac and Majorana spinors you might also like to refer to a comparable type (but much more expert) answer comparing Weyl with Dirac and Majorana spinors.
Meanwhile, as others can probably provide a better answer, I regard this as an opportunity to learn more by inviting corrections; otherwise, I might post a more detailed answer, with references, later.
UPDATE: Abstract of the ref given as this paper in Physics StackExch 381625 states:
"...The physical observables in Schrödinger–Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit." (behind a paywall).