When deriving the (1/2,0) and (0,1/2) representations of the Lorentz group one usually starts by describing how points in Minkowski space transform while preserving the speed of light (or the metric).
This all makes sense. Then one encodes this symmetry into the Lie algebra $so(1,3)$ Lie algebra which turns out to be the same for $su(2) \oplus su(2)$. From all of this we can show that $SU(2) \times SU(2)$ (actually $SL(2,C)$) is the double cover of the Lorentz group. Why this extra degree of freedom is needed or what it represents physically is a question on it's own but that's beside the point.
Now finally one finally identifies Cardan and Casimir elements and derives the Spinor representation of the Lorentz Algebra. The Boost and Rotation generators emerging from the $SL(2,C)$ elements in the spinor repr. then are still called Lorentz transformations.
My Question: During all of this I got completely lost how we can still call these generators, Lorentz generators. What do they have to do with Boosts and Rotations in Minkowski space. Spinors live in a completely different universe. What is the (intuitive) connection between the Boosts/Rotations in Minkowski space to the "same" in Spinor space? Why does a spinor (whatever that is) change when we transform real space?
