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Let $K$ be a field of characteristic different from 2, and let $q$ be a non-degenerate quadratic form on $V=K^n$ for $n\ge 5$, say, $$q(x_1,\dots, x_n)=a_1x_1^2+\dots+a_nx_n^2.$$

Question. How can one describe all maximal tori in the special orthogonal group $G={\rm SO}(V,q)\subset {\rm SL}(V)$?

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    $\begingroup$ There is a paper by Kariyama, On conjugacy classes of maximal tori in classical groups. I don't know off the top of my head which form(s) of the orthogonal group it handles. $\endgroup$ Commented Mar 21 at 16:13
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    $\begingroup$ Kariyama assume that $G$ is quasi-split and that $K$ is a non-archimedean local field. I am interested in the case where $K$ is not local or global! $\endgroup$ Commented Mar 21 at 16:20
  • $\begingroup$ Re, I believe that the introduction mentions local fields as motivation, but that §1.2 only assumes that $K$ is an infinite field of characteristic $\ne 2$. (But I see that you are right that it does assume that $G$ is quasi-split.) $\endgroup$ Commented Mar 21 at 16:40
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    $\begingroup$ @LSpice: In 1.2, on page 134, line $-5$, the author writes: "In the orthogonal case, we furthermore assume $K$ is a non-archimedean local field." $\endgroup$ Commented Mar 21 at 17:17
  • $\begingroup$ Re, I see, you are right. It says that this is to allow them to use the results of [7], Milnor - On isometries of inner-product spaces. I wonder if the 35 years since have offered any improved methods, or if it is just known that the results are false? $\endgroup$ Commented Mar 21 at 17:54

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