Real Classes of Finite Special Unitary Groups

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We classify all real and strongly real classes of the finite special unitary group SUₙ(q). Unless q ≡ 3 (mod 4) and n |4, the classification of real classes is similar to that of the finite special linear group SLₙ(q). We relate strong reality in SUₙ(q) to strong reality in the finite special orthogonal groups SO±ₙ(q), and we classify real and strongly real classes in the last case for the group SO±ₙ(q), when q is odd and n ≡ 2 (mod 4).