Let $G$ be a compact simple Lie group. Choose a system of positive roots, and let $\mathrm{SU}(2) \subset G$ correspond to the highest root, and $\mathbb{Z}/2 \subset \mathrm{SU}(2)$ the centre. The centralizer of this $\mathbb{Z}/2$ inside of $G$ is a subgroup $H \subset G$ of shape $\mathrm{SU}(2) \circ K = (\mathrm{SU}(2) \times K) / (\mathbb{Z}/2)$. The Dynkin diagram for $H$ can be found by drawing the affine dynkin diagram for $G$, and deleting the node(s) adjacent to the affine root. The now-isolated affine root is the copy of $\mathrm{SU}(2)$, and the rest of the Dynkin diagram for $K$. In the type-A case, $H$ is reductive but not simple, picking up a $\mathrm{U}(1)$ factor; this is because in that case the affine root had two neighbours, not just one. The list of $H$'s is available in the Wikipedia article [Quaternion-Kähler symmetric space](https://en.wikipedia.org/wiki/Quaternion-K%C3%A4hler_symmetric_space), because the quotient spaces $G/H$ are precisely the *quaternionic symmetric spaces*.


Standard arguments then say that $G$ has a real form with maximal compact $H$. It is not the compact form (except for $G = H = \mathrm{SU}(2)$), and it is usually not the split real form. Rather, it is a third canonical real form for any group. For the classical series, it is $\mathrm{SU}(2,n-2)$, $\mathrm{SO}(4,n-4)$, and $\mathrm{Sp}(1,n-1)$. If I am reading Wikipedia correctly, then, together with $\mathrm{SO}(3,n-3)$, these are the real forms that admit quaternionic discrete series representations.



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> Does this canonical real form have a standard name in the literature?
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What is the name of the real form corresponding to the quaternionic symmetric space?
In the Crelle paper by Gross and Wallach “On quaternionic discrete series representations, and their continuations” (*J. Reine Andgew. Math.* **481** (1996) 73–123, [available here](https://eudml.org/doc/153874)), the form you mention is described in §3 and called “the quaternionic real form”.


There are other references to “quaternionic real form” [found by Google](https://www.google.com/search?q=%22quaternionic%20real%20form%22), and all seem to refer to the same thing. I have seen it regularly at least to denote the form of $E\_8$ with Cartan index $-24$. So I think we can say that “quaternionic real form” is a reasonably standard term.


(There is still at least some possibility of confusion as [this question](https://mathoverflow.net/questions/299942/real-automorphisms-of-the-quaternionic-real-group-rm-so4m) uses the term, albeit with quotes, to denote a different real form of the $D\_n$ series.)


