This paper examines a systematic method of constructing a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of nonstandard geometric representations. This method can be employed to construct generalizations of root systems for a large family of linear groups generated by involutions. We then give a characterization of Coxeter groups, among these groups, in terms of such paired root systems. Furthermore, we use this method to construct and study the paired root systems for reflection subgroups of Coxeter groups.

We define in an axiomatic fashion a Coxeter datum for an arbitrary Coxeter group $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear pairing without any integrality or nondegeneracy requirements. These representations are not required to be embeddings of $W$ in the orthogonal group of any vector space, and they give rise to a pair of inter-related root systems generalizing the classical root systems of Coxeter groups. We obtain comparison results between these nonorthogonal root systems and the classical root systems. Further, we study the equivalent of the Tits cone in these nonorthogonal representations.