Journal of Lie Theory

A first characterization of the isomorphism classes of k-involutions for any reductive algebraic group defined over a perfect field was given by A. G. Helminck [On the classification of k-involutions I, Adv. in Math. 153 (2000) 1--117] using $3$ invariants. In another paper by A. G. Helminck, L. Wu and C. Dometrius [Involutions of Sl(n, k), (n > 2), Acta Appl. Math. 90 (2006) 91--119] a full classification of all k-involutions on SL(n,k) for k algebraically closed, the real numbers, the p-adic numbers or a finite field was provided. In a paper by R. W. Benim, A. G. Helminck and F. Jackson Ward [Isomorphy classes of involutions of Sp(2n,k), n>2, J. of Lie Theory 25 (2015) 903--947] a similar classification was given for all k-involutions of SP(2n,k).
In this paper, we find analogous results to develop a detailed characterization of the k-involutions of SO(n,k,β), where β is any non-degenerate symmetric bilinear form and k is any field not of characteristic 2. We use these results to characterize the isomorphy classes of k-involutions of SO(n,k,β) for all bilinear forms, β when char(k) is not equal to 2 or 3, and for some bilinear forms when char(k) = 3. When n unequal 3, 4, 6, 8, then the characterization considers all involutions. If n = 3, 4, 6, 8, then the characterization only considers inner involutions.