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Mar 11, 2021 at 16:46 comment added paul garrett @LSpice, well, both $G$-orbit and isomorphism class without necessarily mentioning $G$-orbits... Many possible logical orderings... (And, yes, haha, just like Johny Carson's mythical "Dr. Al Bendova")
Mar 11, 2021 at 0:20 comment added LSpice May I also make a suggestion that looking at reductive groups from both ends must indicate that you have been undertaking a careful study of endoscopy? :-)
Mar 11, 2021 at 0:10 comment added LSpice When you say that $D = (W \oplus W')^\perp$ is independent of choices, you really mean that its $G$-orbit is well determined, right? From the 'intrinsic' point of view, I think of this as the conjugacy of maximal split tori $S$, with the isometry/similitude group of $D$ being the centraliser of $S$. But maybe I missed your point about what you were seeking to describe intrinsically (or about what counts as intrinsic).
Mar 10, 2021 at 23:49 history edited paul garrett CC BY-SA 4.0
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Mar 10, 2021 at 23:43 history edited paul garrett CC BY-SA 4.0
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Mar 10, 2021 at 23:29 history answered paul garrett CC BY-SA 4.0