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**SU(2) a double cover for Lorentz group?**

I'm presently reading the new book, "Symmetry and the Standard Model", by Matthew Robinson. On page 120, he writes, "the Lorentz group (SO(1,3), pg 117) is actually made up of two copies of SU(2). We want to reiterate that this is only true in 1+3 spacetime dimensions." I suppose this is not a surprise since they started with the 1+3 version of special relativistic QFT to derive the SU(2) group to begin with.

But suppose we had some formulism that necessitated the internal SU(2) structure before imposing spacetime coordinates. If we found "two copies of SU(2)" in this formulism, would that necessitate the existence of a SO(1,3) Lorentz group on any underlying spacetime? So I guess I'm asking about internal verses external symmetries, about local verses global symmetries, and about how to connect internal SU(2) symmetry with spacetime SO(1,3) symmetries. At this point I don't really know how "two copies of SU(2)" "cover" SO(1,3). If anyone had a summary or a broad outline of how this all works, I'd appreciate it. If I can understand this, then perhaps I could understand how that one formulism could derive both QM and GR. Thanks.