Is there a way to explicitly compute the homology of the space $$ \varinjlim_{(p,q)} SO(p,q)^+, $$ where each $SO(p,q)$ is the indefinite special orthogonal group, and $SO(p,q)^+$ its identity component, where $\mathbb{N}\times\mathbb{N}$ has the product order, and where we have inclusions $SO(p,q)\hookrightarrow SO(p',q')$ defined by $A\mapsto \mathbb{I}_{p'-p}\oplus A\oplus (-\mathbb{I}_{q'-q})$?

What I tried to argue is:

- The maximal connected part of $SO(p,q)^+$ is homotopy equivalent to $SO(p)\times SO(q)$;
- There is the chain of isos $$ \begin{align} \textstyle H_n\left( \varinjlim SO(p,q)^+\right) & \cong \textstyle \varinjlim H_n(SO(p,q)^+) \\ &\cong \textstyle \varinjlim H_n\Big(SO(p)\times SO(q)\Big)\\ &\cong \displaystyle \varinjlim \bigoplus_{a+b=n} H_a(SO(p))\otimes_{\mathbb Z}H_b(SO(q))\oplus \text{ some Tor} \end{align} $$ And this seems kinda tame, up to knowing completely the homology of the two factors. Is there a nifty way to conclude something more?