In physics, both $SO(3)$ and $SO^{+}(3,1)$ Lie groups are of paramount importance. Things start to become very curious when we realize that in Quantum Physics (Non-Relativistic Quantum mechanics and Quantum Field Theory), their double covers, $\mathrm{DC}_{SO(3)} = SU(2)$, $\mathrm{DC}_{SO^{+}(3,1)} = SL(2,\mathbb{C})$ also plays a major role. The physical conclusion is that the representations of the first Lie groups gives us scalar,vector and tensor matter (Higgs bosons, Gauge bosons and Gravitons), while the representations of their double covers gives us spinor matter (electrons, quarks,etc...).
But, what I would like to ask lies on a property that these double covers share: they are isomorphic to Spin groups:
$$SU(2) \approx Spin(3)$$ $$SL(2,\mathbb{C}) \approx Spin(3,1).$$
So,
Given a Lie group $G$ and its double cover $DC_{G}$, is it always the case that we have, automatically, a isomorphic property between $DC_{G}$ and $Spin(G)$: $$DC_{G} \approx Spin(G)?$$