$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$We know that $$ \Spin(1,3)=\SL(2,\mathbb C) $$ and $$ \Spin(4)=\SU(2) \times \SU(2). $$

The $\Spin(1,3)$ is the Lorentz version of Spin group, while $\Spin(4)$ is the Euclidean version of Spin group.

The spinor representations of $\Spin(4)$ is essentially the spinor representation of $\Spin(4)=\SU(2) \times \SU(2)$ thus it is labeled by two components of spinors, each for the spinor representation of $\SU(2) \times$.

How do we understand the spinor representations of $\Spin(1,3)$? Note that $\Spin(1,3)$ is non-compact. How is this $\Spin(1,3)$ spinor related to the spinor of the compact group $\Spin(4)$?

How do we understand the 2 dimensional spinor as complex Weyl and 4 dimensional spinor as complex Dirac but NO 4 dimensional real Majorana in $\Spin(4)$?

It is also said $\Spin(1,3)$ is the

**complexification**of $\SU(2)$. How to see this from spinor representation living in the vector spaces?How do we understand the 2 dimensional spinor as complex Weyl and 4 dimensional spinor as complex Dirac or real Majorana in $\Spin(3,1)$?

See https://en.wikipedia.org/wiki/Spinor#Spinors_in_representation_theory

Refs from google search:

http://www-personal.umich.edu/~williams/notes/spinor.pdf http://scipp.ucsc.edu/~haber/ph251/Spinor_Shijun https://en.wikipedia.org/wiki/Complexification_(Lie_group)