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A first characterization of the isomorphism classes of k-involutions for any reductive algebraic group defined over a perfect field was given in [Hel00] using 3 invariants. In [HWD04, HW02] 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 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 classify the isomorphy classes of k-involutions of SO(n, k, β) for some bilinear forms and some fields k.

In this paper, we consider the order m k-automorphisms of SL(2, k). We first characterize the forms that order m k-automorphisms of SL(2, k) take and then we simple conditions on matrices A and B, involving eigenvalues and the field that the entries of A and B lie in, that are equivalent to isomorphy between the order m k-automorphisms Inn A and Inn B . We examine the number of isomorphy classes and conclude with examples for selected fields.

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