We give a simple proof of the known fact that the symplectic quadrangle is self-dual if and only if the ground field is perfect of characteristic~2, and that a polarity exists exactly if there is a root of the Frobenius automorphism. Moreover, we determine all polarities, characterize the conjugacy classes of polarities, and use the results to give a simple proof that the centralizer of any polarity acts two-transitively on the ovoid of absolute points. The proofs use elementary calculations in solvable subgroups of the symplectic group.
"Polarities of Symplectic Quadrangles." Bull. Belg. Math. Soc. Simon Stevin 10 (3) 437 - 449, September 2003. https://doi.org/10.36045/bbms/1063372348