Isomorphy Classes of Involutions of SO(n, k, β), n>2

A first characterization of the isomorphism classes of k -involutions for any reductive algebraic group defined over a perfect field was given in On the Classification of k -involutions I using 3 invariants.In Involutions of SL(n, k) , (n > 2) 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 Isomorphy Classes of Involutions of SP(2n, k) , (n > 2) , 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, ) , where 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, ) for all bilinear forms, when char(k) = 2, 3 , and for some bilinear forms when char(k) = 3 .When n = 3, 4, 6, 8 , then the characterization considers all involutions.If n = 3, 4, 6, 8 , then the characterization only considers inner involutions.