  In this paper, we generalize the formal affine Demazure algebra of
Hoffnung-Malag\'on-L\'opez-Savage-Zainoulline to all real finite reflection
groups. We begin by generalizing the formal group ring of
Calm\`es-Petrov-Zainoulline to all real finite reflection groups. We then
define and study the formal Demazure operators that act on the formal group
ring. Using these results and constructions, we define and study the formal
affine Demazure algebra for all real finite reflection groups. Finally, we
compute several structure coefficients that appear in a braid relation among
the formal Demazure elements, and we conclude this paper by computing all
structure coefficients for the reflection groups $I_2(5)$, $I_2(7)$, $H_3$, and
$H_4$.
