A Factorization Result for Classical and Similitude Groups

For most classical and similitude groups, we show that each element can be written as a product of two transformations that preserve or almost preserve the underlying form and whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of Meeglin, Vigneras, and Waldspurger on the existence of automorphisms of p-adic classical groups that take each irreducible smooth representation to its dual.