Starting with the double cover of the conformal group of Minkowski space, $SO_0(2,4)$, let $\mathfrak{M}^{AB}$ be the generators (basis elements) of the corresponding Lie algebra $\mathfrak{so}(2,4)$ where $A,B = 0,...,5$. I have identified a Lie subalgebra generated by the elements
$$ P_-: = P_1 - P_2 = \mathfrak{M}^{10}+ \mathfrak{M}^{15} - \mathfrak{M}^{20}- \mathfrak{M}^{25} $$
$$ K_+: = K_1 + K_2 = \mathfrak{M}^{10}- \mathfrak{M}^{15} + \mathfrak{M}^{20}- \mathfrak{M}^{25} $$
$$ M^3 : = iH + M^{12} = \mathfrak{M}^{05} - \mathfrak{M}^{12} $$
with commutation relations $[P_-, K_+] = - 4 i M^3$, $[P_-, M^3] = - 2 i P_-$, $[K_+, M^3] = 2 i K_+$. This algebra is isomorphic over the reals to $\mathfrak{so}(2,1, \Bbb{R}) \simeq \mathfrak{sl}(2, \Bbb{R}) \simeq \mathfrak{su}(1,1)$ via the mapping $\tilde{M}^{01} = - \frac{1}{2} (P_- + K_+), \tilde{M}^{02} = - \frac{1}{2}(P_- - K_+), \tilde{M}^{12}= \frac{1}{2}M^3$ where $\tilde{M}^{ij}$ are the generators of $\mathfrak{so}(2,1, \Bbb{R})$.
I want to be able to say something about the global topology of the Lie subgroup generated by $\{P_-, K_+, M^3\}$ and therefore I would like to identify the Lie subgroup it generates. My question is thus: how can I identify the Lie subgroup generated by $\{P_-, K_+, M^3\}$?
It appears the answer could be either $SO(2,1)$ or $SL(2, \Bbb{R}) \simeq SU(1,1)$, which have different topologies (the former being non-compact and not connected while the latter is non-compact and connected) but I do not know of how to distinguish between these given the information I have. My hope was there is some constraint coming from the topology of the ambient group $SO_0(2,4)$ which is connected and non-compact [1]. Along this line of thinking I know of the standard Lie subgroup-subalgebra theorems found in [2] describing the existence of a unique connected Lie subgroup to every Lie subalgebra. However, I don't believe this helps me (it actually confuses me because $SO_0(2,1)$ and $SL(2, \Bbb{R})$ are both connected but not isomorphic).
Finally, if there is no way to identify the Lie subgroup from the given information, and it is the case that I need some more global information about the Lie subgroup, what information could that be? Is there a calculation I could do or theorem I could use? Perhaps there is a way to move around on the submanifold which represents the Lie subgroup corresponding to $\{P_-, K_+, M^3\}$ and calculate the curvature or some global characteristic to distinguish between the Lie subgroups?