# Is there an easy way to know the covering group of $O(p,q)$ ?

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I would like to know how to find the double cover of say $SO(2,4)$ All references say that it is $SU(2,2)$ Why not $SU(4)$ ? There is something still missing in my understanding.

asked Feb 14, 2018
reshown Feb 15, 2018

Indeed, we should have Spin(2,4) = Spin(6) since Cl(2,4) = Cl(6). I must be missing something too.

I know that these clifford algebras don't match. I might be stupid but I wanted to know whether we can know this quickly from analogy with something like $SL(2,C)=SO(1,3)$ like may be try to find matrices living in SU or SL . Can you get confused reasoning about whether the right covering group of it $SU(1,3)$ or $SU(2.2)$ ?

I think in principle the best approach will be to understand how to go from the Clifford algebra ( which is either a matrix algebra over R, C, or H or a sum of two matrix algebras:  https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras#Real_case ) to its group of units. I'm not sure how the Clifford norm appears in these matrix representations though.

Nevermind, I was briefly confused about centers. The center of SU(p,q) is always Z/p+q.

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