string theory - Lorentz spinors of $SO(n,1)$ and conformal spinors of $SO(n,2)$

• It would be great if someone can give me a reference (short enough!) which explains the (spinor) representation theory of the groups $SO(n,1)$ and $SO(n,2)$.

I have searched through a few standard representation theory books and I couldn't find any.

• More specifically I would like to know how a Lorentz spinor of $SO(n-1,1)$ (say $Q$) is "completed" to a conformal spinor of $SO(n,2)$ (say $V$) by saying,

$V = (Q, C\bar{S})$

where $C$ is a "charge conjugation operator" and $S$ is probably another $SO(n-1,1)$ spinor.

Is there some natural Clifford algebra representation ($\Gamma$) lurking around here with respect to which I can define the "charge conjugation operator" as $C$ such that $C^{-1}\Gamma C = - \Gamma ^T$? (...in general a representation of the Clifford algebra also gives a representation of $SO(n,1)$..I would like to know as to how this general idea might be working here...)

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  Some of the other aspects of this group theory that I want to know are an explanation for facts like,

  
  
  
  
  
  • $Sp(4)$ is the same as $SO(3,2)$, and the fundamental of $Sp(4)$ is the spinor of $SO(3,2)$
  
  
  • $SU(2,2)$ is the same as SO(4,2), and the fundamental of $SU(2,2)$ is the spinor of $SO(4,2)$
  
  
  
  

  (...just two "facts" hoping that people can point me to some literature (hopefully short!) which explains the systematics of which the above are probably two examples...)