I recently asked for some references regarding the representation theory of $SO(n,m)$ (specifically for the case $n,m>2$), but unfortunately it is looking like I won't have much luck in that regard. Thus, I figure I should just work on some simpler cases myself... but I ran into a problem due to my lack of algebraic topology skills.
What I know so far: Consider the case of, say, $SO(2,4)$, and even more specifically let's look at the identity component of it, $SO^+(2,4)$. The maximal compact subgroup of this component is $SO(2)\times SO(4)$, and so one has that
$$\pi_1(SO^+(2,4))\cong \pi_1(SO(2))\times\pi_1(SO(4))\cong\mathbb{Z}\times\mathbb{Z}_2.$$
I also know that $\text{Spin}(2,4)$ is a double cover of $SO^+(2,4)$, and there is some relationship between the fundamental group of a space and the fundamental group of its double cover.
Some questions:
(1) What, precisely, is the relationship between $\pi_1(SO^+(2,4))$ and $\pi_1(\text{Spin}(2,4))$?
EDIT: In regards to (1), I'm fairly certain that one should have
$$\pi_1(\text{Spin}(2,4))<\pi_1(SO^+(2,4)),$$
and moreover that the index $[\pi_1(\text{Spin}(2,4)):\pi_1(SO^+(2,4))]$ should be 2 (since it is a double cover). Is that correct? Also, wouldn't that mean $\pi_1(\text{Spin}(2,4))$ is either $\mathbb{Z}\times\{0\}$ or $2\mathbb{Z}\times\mathbb{Z}_2$? Or are there other possibilities?
(2) More importantly, given that I'm pretty sure the answer to the above question will reveal that $\text{Spin}(2,4)$ is not simply connected, what is the universal cover for $SO^+(2,4)$? How can I go about determining it? As I said, I'm a bit lacking in the algebraic topology area, so I'm not really even sure where to start -- any help would be tremendously appreciated.