Let X be an orientable smooth manifold and let LX be its frame bundle. It is well-known that the global section of the quotient bundle between the frame bundle LX (with structure group GL$^+$(4)) and the SO(3,1) group are tetrad fields. The tetrad fields uniquely determine a metric. The connections are the metric connections.
I am interested in the spin version of this concept.
In a previous question I was told that I cannot take LX/Spin(3,1) because Spin is not in GL$^+$. Ok, fine.
However, some reading lead me across a concept known as a double cover of GL$^+$(4) (noted $\widetilde{{\rm GL}^+}(4)$) leading to a double cover of the frame bundle $\widetilde{LX}$.
Can one then take $\widetilde{LX}/Spin(3,1)$ and obtain a tetrad field, where the connection is a spin connection?
I am also interested in learning a bit more about $\widetilde{LX}$ and how can I define its elements canonically. I known that GL$^+$(4) can be represented as the exponentials of matrices. I also know that Spin relates to the exponentials of bivectors in Clifford algebra. From this, can I intuit that the expression of $\widetilde{{\rm GL}^+}(4)$ are the exponentials of the multivectors of the Clifford algebra?