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"In this article, we begin a theory linking hyperplane arrangements and invariant forms for reflection groups over arbitrary fields.... Let V be an n -dimensional vector space over a field F, and let G ≤ Gl n (F) be a finite group.... An element of finite order in Gl( V ) is a reflection if its fixed point space in V is a hyperplane, called the reflecting hyperplane . There are two types of reflections: the diagonalizable reflections in Gl( V ) have a single nonidentity eigenvalue which is a root of unity
-- Julia Hartmann and Anne V. Shepler, " Reflection Groups and Differential Forms ," Mathematical Research Letters , Vol. 14, No. 6 (Nov. 2007), pp. 955-971
"… the class of reflections is larger in some sense over an arbitrary field than over a characteristic zero field. The reflections in Gl( V ) not only include diagonalizable reflections (with a single nonidentity eigenvalue), but also transvections , reflections with determinant 1 which can not be diagonalized. The transvections in Gl( V ) prevent one from developing a theory of reflection groups mirroring that for Coxeter groups or complex reflection groups."
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