  Let K be a field and G a split connected reductive affine algebraic K-group.
Let T be a split maximal torus of G, W its finite Weyl group, and R its root
system. After fixing a realization of R in G and choosing a simple system for
R, one gets a system of representatives for W in G(K), called the Canonical
Representatives. It is well-known that these representatives rarely form a
subgroup, and it is necessary for some questions to understand and quantify
this failure. Various new formulas are given which constitute progress in this
direction. An application of such formulas to the simple supercuspidals of
Gross-Reeder and Reeder-Yu is provided.
