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?

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    $\begingroup$ This is a fairly trivial answer, but the term “quaternionic real form” is found by Google in a number of texts and seems to correspond to what you're describing. (Thanks for pointing out that this form is canonical, I had never realized this!) $\endgroup$ – Gro-Tsen May 3 at 17:20

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.)

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  • $\begingroup$ Thanks. It feels like a good name. But I didn't know if it had another one. $\endgroup$ – Theo Johnson-Freyd May 3 at 18:40
  • $\begingroup$ Actually, I don't know which form the linked question is referring to. I should be able to divine it: it is a form of Spin(4n) in which one of the half-spin representations is real and the other is quaternionic. $\endgroup$ – Theo Johnson-Freyd May 3 at 18:44
  • $\begingroup$ @TheoJohnson-Freyd Regarding the form mentioned in the linked question (which, again, is not the one you were asking about), the notation $SO^*(n)$ is fairly standard, and it's the only real form of $SO(n)$ (for $n$ even) that's not a $SO(k,n-k)$. Its maximal compact subgroup is $U_n$ and it's called DIII in Cartan's notation. $\endgroup$ – Gro-Tsen May 3 at 19:17
  • $\begingroup$ Thanks! I do know that real form, I just didn't know its name, and I don't immediately see why it would be called "quaternionic". $\endgroup$ – Theo Johnson-Freyd May 4 at 22:26

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