Considering The moduli space of $K3$ surface, $\frac{SO(4,20)}{SO(4)\times SO(20)}$ , I would like to get a split of the form : $\frac{SO(4,20)}{SO(4)\times SO(20)}$ = $SO(4,4)$ + another part to determine.
We have theorem for general $SO(p,q)$ that says it can split if $p=q $, and quasi-split if $|p-q| \le 2$, but it doesn't say much for what I am looking for.
I wonder if such decomposition is possible, and if it is so how to do it? A link or refs to attack the subject would be nice.