TITLE: What is the name of the real form corresponding to the quaternionic symmetric space?
QUESTION [5 upvotes]: 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, 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.

Does this canonical real form have a standard name in the literature?

REPLY [2 votes]: 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), 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, 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 uses the term, albeit with quotes, to denote a different real form of the $D_n$ series.)