I know that the distinction of real / complex Lie algebra.
For example,
The $su(2)=so(3)=sp(1)$ is a real Lie algebra.
The $sl(2,\mathbf{C})=so(1,3)$ is a complex Lie algebra.
Question 1: How are their representations, being real / pseudoreal/ complex representations, related to the fact whether their Lie algebras are real / complex Lie algebra or not?
The $su(2)$ fundamental rep is pseudoreal. It is a 2-dimensional rep of $su(2)$.
The $sl(2,\mathbf{C})$ fundamental rep is complex. It is a (2,0) or (0,2)-dimensional rep of $su(2)$.
Question 2: We can consider the spin group. Note that all spin groups has real Lie algebras. Such that
$Spin(3)=SU(2)$ has its spinor rep is pseudoreal.
$Spin(4)=SU(2)\times SU(2)$ has its spinor rep is pseudoreal.
$Spin(5)$ has its spinor rep is pseudoreal.
$Spin(6)$ has its spinor rep is complex.
$Spin(7)$ has its spinor rep is real.
$Spin(8)$ has its spinor rep is real.
$Spin(9)$ has its spinor rep is real.
$Spin(10)$ has its spinor rep is complex.
And there is a modular 8 Bott periodicity.
So how are the real / complex Lie algebra $\Rightarrow$ determine Real / pseudoreal/ complex representations? Maybe I have missed some important ingredients. Please spell out and clarify the conceptual relations between the twos.