© The Author 2014. A special linear Grassmann variety SGr(k, n) is the complement to the zero section of the determinant of the tautological vector bundle over Gr(k, n). For an SL-oriented representable ring cohomology theory A∗(-) with invertible stable Hopf map η, including Witt groups and MSLη∗,∗, we have A∗(SGr(2, 2n + 1)) ≅ A∗(pt)[e]/(e2n), and A∗(SGr(k, n)) is a truncated polynomial algebra over A∗(pt) whenever k(n - k) is even. A splitting principle for such theories is established. Using the computations for the special linear Grassmann varieties, we obtain a description of A∗(BSLn) in terms of homogeneous power series in certain characteristic classes of tautological bundles.