The ${\rm Spin}^c(3,1)$ group is used in physics. From the references1 I found that
$$ {\rm Spin}^c={\rm Spin}(V) \times S^1 $$
But I was not able to find any references for its lie algebra. Specifically, I am interested in the ${\rm Spin}^c(3,1)$ case. What is its lie algebra?
Furthermore, this page2 mentions that ${\rm Spin}^c(n)$ is used in physics, but does not confirm that it is ${\rm Spin}^c(3,1)$ specifically that is used in physics. I suspect that it is, but it would be nice to confirm that, also.
My attempt:
I know that $S^1$ is isomorphic to $U(1)$. So would $\mathfrak{so}(3,1) \oplus i$ be an isomorphic representation of ${\rm Spin}^c(3,1)$?