I think this should have an answer, but I can't see what it is. It's inspired by the section labelled "Spinors" in Parker's and Taubes's paper, "On Witten's Proof of the Positive Energy Theorem." Here's the question:
Take $V$ be a real four dimensional vector space with an inner product of signature $(3,1)$, e.g. Minkowski space. By choosing a timelike vector $e_0$ we get an inclusion of $SO(3)$ into the identity component of $SO(3,1)$ by identifying elements of $SO(3)$ with transformations that fix the the orthogonal complement of $e_0$. Does this induce an injection of $Spin(3)$ into $Spin(3,1)$ such that the inclusions commute with the covering maps?
Yes it does; in fact, I think any morphism $SO(u,v) \rightarrow SO(u+i,v+j)$ induced by an inclusion along those lines is going to have such a map. It should be easier to see this topologically, though; any circle induced by an inclusion from $SO(2)$ is going to generate the fundamental group element, and those can be embedded in the smaller group easy.
Algebraically, note that $Spin(3) \simeq SU(2)$ and $Spin(3,1) \simeq SL(2,\mathbf{C})$ and the inclusion on the $SU/SL$ level is found via Pauli spin matrices.