$Spin(6,2) = SU(2, 2, \mathbb{H})$

Question

On Wikipedia https://en.wikipedia.org/wiki/Spin_group#Indefinite_signature, it says $Spin(6,2) = SU(2, 2, \mathbb{H})$. But I cannot find any reference. Does anyone know one, or any other references which explains what this spin group is? Thank you in advance!

Answer 1

This is a case that shows what a great thing the Tits index a.k.a. Satake diagram is. Here is Remark 4.5.20.ii from my thesis:

(The reference Jac6 is Lie Algebras by Nathan Jacobson. Interscience Tracts in Pure and Applied Mathematics, Number 10. John Wiley and Son, Inc. New York, 1962.)

The algebra belonging to the thing on the right would almost always be written $\mathfrak{so}(6,2)$, and then of course its corresponding simply connected group is $Spin(6,2)$. The thing on the left, unfortunately, seems to not have a standard name (neither the Lie algebra nor its corresponding simply connected group), so in every source we have to check the nomenclature. In Schulte-Hengesbach's diploma thesis mentioned by Dietrich Burde in a comment, the Lie algebra is called $\mathfrak{sp}(2, \mathbb H)$; in Onishchik/Vinberg's tables, the Lie algebra is called $\mathfrak{u}^*_{4}(\mathbb H)$; apparently your source calls the corresponding simply connected group $SU(2,2, \mathbb H)$.

Finally, here is how Tits describes both forms in his article in the Boulder Proceedings (p.56/57). He talks about groups. The form on the left is, in Tits' notation, $^1D^{(2)}_{4,2}$ (i.e. $n=4, d=2, r=2$), and he would call it $SU_4(\mathbb H, h)$ for a certain $h$ as described below (I think Jacobson's and my description above matches rather his "equivalent description for $d \ge 2$", but I'm not entirely sure about that). The form on the right is $^1D^{(1)}_{4,2}$ (i.e. $n=4, d=1, r=2$).